Introduction
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Categorization of Variables and Measuring Scales
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Summation
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Measure of Central Tendency
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Measures of Dispersion - Range, Variance, and Standard Deviation
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Measures of Dispersion - Interquartile Range
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"2m ","ChapterTopicVideoID":12809,"CourseChapterTopicPlaylistID":265038,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12809.jpeg","UploadDate":"2018-07-30T17:08:05.7770000","DurationForVideoObject":"PT2M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"This chapter, I\u0027ll be talking about descriptive statistics."},{"Start":"00:03.180 ","End":"00:05.760","Text":"Specifically, I\u0027ll introduce the subject,"},{"Start":"00:05.760 ","End":"00:10.980","Text":"since this will be the 1st in a series of lectures dealing in descriptive statistics."},{"Start":"00:10.980 ","End":"00:12.990","Text":"In descriptive statistics,"},{"Start":"00:12.990 ","End":"00:14.730","Text":"we study sets of things."},{"Start":"00:14.730 ","End":"00:17.895","Text":"A set can be anything and everything that we want."},{"Start":"00:17.895 ","End":"00:20.955","Text":"A set may contain a group of,"},{"Start":"00:20.955 ","End":"00:23.580","Text":"let\u0027s say, kindergarten children, for example,"},{"Start":"00:23.580 ","End":"00:26.310","Text":"a group of shares in a portfolio,"},{"Start":"00:26.310 ","End":"00:28.245","Text":"all the residents in a given city,"},{"Start":"00:28.245 ","End":"00:30.330","Text":"basically anything and everything as I said,"},{"Start":"00:30.330 ","End":"00:33.090","Text":"and each group has its members."},{"Start":"00:33.090 ","End":"00:37.920","Text":"Each member is basically an individual within the group."},{"Start":"00:37.920 ","End":"00:41.410","Text":"For example, in a group of kindergarten children well,"},{"Start":"00:41.410 ","End":"00:43.100","Text":"Joe and Karen,"},{"Start":"00:43.100 ","End":"00:47.010","Text":"and Jane are individual children here."},{"Start":"00:47.010 ","End":"00:51.770","Text":"They\u0027re members in the group and a group of shares in a portfolio, well,"},{"Start":"00:51.770 ","End":"00:59.795","Text":"it could be shares from Microsoft or Oracle or any industry whatsoever."},{"Start":"00:59.795 ","End":"01:06.265","Text":"Each share is a member in the group of shares in a specific portfolio,"},{"Start":"01:06.265 ","End":"01:08.790","Text":"all the residents in a given city."},{"Start":"01:08.790 ","End":"01:12.530","Text":"Well again, those are the individuals living in New York."},{"Start":"01:12.530 ","End":"01:17.360","Text":"Well each individual living in New York is a member of this group."},{"Start":"01:17.360 ","End":"01:19.580","Text":"Now, amongst the members of a group,"},{"Start":"01:19.580 ","End":"01:22.680","Text":"there are factors that may receive a number of values."},{"Start":"01:22.680 ","End":"01:25.370","Text":"Now these factors are called variables."},{"Start":"01:25.370 ","End":"01:30.630","Text":"For example, the daily return on a share in a portfolio,"},{"Start":"01:30.630 ","End":"01:32.255","Text":"well, that\u0027s a variable."},{"Start":"01:32.255 ","End":"01:36.140","Text":"The length of time the share has been held, that\u0027s another variable."},{"Start":"01:36.140 ","End":"01:39.170","Text":"The industry sector the share belongs to,"},{"Start":"01:39.170 ","End":"01:41.750","Text":"again the 3rd variable and so on and so forth."},{"Start":"01:41.750 ","End":"01:43.495","Text":"Now, all of these variables,"},{"Start":"01:43.495 ","End":"01:46.680","Text":"all right, they vary from 1 share to another."},{"Start":"01:46.680 ","End":"01:50.390","Text":"In descriptive statistics, we look at a specific group."},{"Start":"01:50.390 ","End":"01:54.575","Text":"Within that group, we collect data about its variables"},{"Start":"01:54.575 ","End":"01:59.850","Text":"and learn how to present and analyze various aspects of that data."}],"ID":13288},{"Watched":false,"Name":"Example","Duration":"7m 31s","ChapterTopicVideoID":12810,"CourseChapterTopicPlaylistID":265038,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12810.jpeg","UploadDate":"2018-07-30T17:08:40.1670000","DurationForVideoObject":"PT7M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.205","Text":"Let\u0027s take a look at an example."},{"Start":"00:02.205 ","End":"00:04.645","Text":"A portfolio has 10 shares."},{"Start":"00:04.645 ","End":"00:06.930","Text":"The portfolio manager publishes"},{"Start":"00:06.930 ","End":"00:10.395","Text":"the return on each share in a specific year and we\u0027re asked,"},{"Start":"00:10.395 ","End":"00:13.035","Text":"what is the group being researched?"},{"Start":"00:13.035 ","End":"00:16.665","Text":"Well, the group here is the group of shares."},{"Start":"00:16.665 ","End":"00:19.185","Text":"What\u0027s the size of the group?"},{"Start":"00:19.185 ","End":"00:21.165","Text":"Well, we\u0027re given that here,"},{"Start":"00:21.165 ","End":"00:23.260","Text":"n, that\u0027s the size,"},{"Start":"00:23.260 ","End":"00:24.980","Text":"that equals to 10 shares."},{"Start":"00:24.980 ","End":"00:27.125","Text":"We have 10 shares in the portfolio."},{"Start":"00:27.125 ","End":"00:30.799","Text":"Then we\u0027re asked, what\u0027s the variable being researched?"},{"Start":"00:30.799 ","End":"00:33.310","Text":"Well, we denote the variable as X."},{"Start":"00:33.310 ","End":"00:37.970","Text":"We have to ask, what varies from share to share?"},{"Start":"00:37.970 ","End":"00:42.140","Text":"Well, the return on each share in a specific year."},{"Start":"00:42.140 ","End":"00:50.580","Text":"So X would be the return in a specific year for each share."},{"Start":"00:51.190 ","End":"00:56.075","Text":"Let\u0027s take a look now at the types of variables that we\u0027re going to have to deal with."},{"Start":"00:56.075 ","End":"00:58.040","Text":"Well, there are 2 types of variables,"},{"Start":"00:58.040 ","End":"01:01.615","Text":"1 is qualitative and the other is quantitative."},{"Start":"01:01.615 ","End":"01:04.044","Text":"Within the quantitative variables,"},{"Start":"01:04.044 ","End":"01:05.320","Text":"we have 2 categories."},{"Start":"01:05.320 ","End":"01:08.505","Text":"We have discrete variables and continuous variables."},{"Start":"01:08.505 ","End":"01:12.165","Text":"Let\u0027s define each 1 of these categories,"},{"Start":"01:12.165 ","End":"01:14.160","Text":"each 1 of these types of variables."},{"Start":"01:14.160 ","End":"01:18.560","Text":"We\u0027ll start with the qualitative and quantitative categories."},{"Start":"01:18.680 ","End":"01:24.965","Text":"Qualitative variables are variables whose values have no meaning of more or less."},{"Start":"01:24.965 ","End":"01:28.540","Text":"The values of these variables have no quantitative content."},{"Start":"01:28.540 ","End":"01:31.530","Text":"For example, a person\u0027s place of residence."},{"Start":"01:31.530 ","End":"01:33.100","Text":"A person can be living in New York,"},{"Start":"01:33.100 ","End":"01:35.890","Text":"Chicago, Los Angeles, any city whatsoever."},{"Start":"01:35.890 ","End":"01:38.365","Text":"If we have 2 people, for example,"},{"Start":"01:38.365 ","End":"01:44.405","Text":"we can\u0027t say that 1 lives more in New York than the other lives in Chicago."},{"Start":"01:44.405 ","End":"01:46.385","Text":"Or for another example,"},{"Start":"01:46.385 ","End":"01:47.495","Text":"a person\u0027s gender,"},{"Start":"01:47.495 ","End":"01:49.190","Text":"could be male or female."},{"Start":"01:49.190 ","End":"01:54.140","Text":"Now, can you say that 1 person is more male than the other is female?"},{"Start":"01:54.140 ","End":"01:56.720","Text":"You can\u0027t, it doesn\u0027t make any sense."},{"Start":"01:56.720 ","End":"01:58.160","Text":"Or a third example,"},{"Start":"01:58.160 ","End":"01:59.660","Text":"a person\u0027s family status,"},{"Start":"01:59.660 ","End":"02:02.240","Text":"could be single, married, divorced, or widower."},{"Start":"02:02.240 ","End":"02:08.855","Text":"Can you say that 1 person is more single than the other is married?"},{"Start":"02:08.855 ","End":"02:12.700","Text":"There\u0027s no sense in this type of sentence."},{"Start":"02:12.700 ","End":"02:17.150","Text":"This is the characteristic of a qualitative variable."},{"Start":"02:17.150 ","End":"02:21.955","Text":"They\u0027re used mostly to categorize the members in the group."},{"Start":"02:21.955 ","End":"02:24.840","Text":"Let\u0027s take a look at quantitative variables now."},{"Start":"02:24.840 ","End":"02:26.960","Text":"These are variables whose values are numbers."},{"Start":"02:26.960 ","End":"02:32.480","Text":"We use numbers as values for these variables and they do have quantitative meanings."},{"Start":"02:32.480 ","End":"02:33.980","Text":"You can take these numbers,"},{"Start":"02:33.980 ","End":"02:37.095","Text":"there is meaning in adding them together,"},{"Start":"02:37.095 ","End":"02:38.840","Text":"multiplying them, dividing them,"},{"Start":"02:38.840 ","End":"02:41.000","Text":"subtracting them, and so on and so forth."},{"Start":"02:41.000 ","End":"02:43.565","Text":"For example, a person\u0027s height in centimeters,"},{"Start":"02:43.565 ","End":"02:48.125","Text":"a person can be taller or shorter than his friend."},{"Start":"02:48.125 ","End":"02:50.650","Text":"Not only that, the difference makes sense."},{"Start":"02:50.650 ","End":"02:56.410","Text":"We\u0027re talking about the difference between 2 people in height."},{"Start":"02:56.410 ","End":"02:59.765","Text":"That has a quantitative content."},{"Start":"02:59.765 ","End":"03:01.175","Text":"There is, here,"},{"Start":"03:01.175 ","End":"03:02.945","Text":"a meaning of more or less."},{"Start":"03:02.945 ","End":"03:07.070","Text":"Another example could be that of a mark on an exam."},{"Start":"03:07.070 ","End":"03:10.370","Text":"1 person can have a better mark than the other person."},{"Start":"03:10.370 ","End":"03:13.850","Text":"Again, there is a meaning of more or less here,"},{"Start":"03:13.850 ","End":"03:19.465","Text":"there is a quantitative content that we can relate to."},{"Start":"03:19.465 ","End":"03:26.480","Text":"Another thing that we have to look at or we can see is that in qualitative variables,"},{"Start":"03:26.480 ","End":"03:31.080","Text":"for example, we have the use of codes."},{"Start":"03:31.080 ","End":"03:34.400","Text":"For example, in a person\u0027s gender right here,"},{"Start":"03:34.400 ","End":"03:42.130","Text":"male could be codified as number 1 and female could be codified as number 2."},{"Start":"03:42.130 ","End":"03:46.565","Text":"We are using numbers here like we are in quantitative variables,"},{"Start":"03:46.565 ","End":"03:52.460","Text":"but again here, 1 and 2 are just a replacement for male and female."},{"Start":"03:52.460 ","End":"03:55.700","Text":"There is no meaning of more or less,"},{"Start":"03:55.700 ","End":"03:57.905","Text":"there is no quantitative content."},{"Start":"03:57.905 ","End":"04:03.420","Text":"You can\u0027t take the value male and multiply it by 2 or the number 1,"},{"Start":"04:03.420 ","End":"04:04.440","Text":"the code of male,"},{"Start":"04:04.440 ","End":"04:05.865","Text":"and multiply that by 2."},{"Start":"04:05.865 ","End":"04:07.730","Text":"2 times 1 has no meaning,"},{"Start":"04:07.730 ","End":"04:10.385","Text":"has no quantitative content."},{"Start":"04:10.385 ","End":"04:16.855","Text":"These are the differences between qualitative and quantitative variables."},{"Start":"04:16.855 ","End":"04:20.710","Text":"We\u0027ll distinguish between 2 types of quantitative variables,"},{"Start":"04:20.710 ","End":"04:23.485","Text":"discrete variables and continuous variables."},{"Start":"04:23.485 ","End":"04:28.630","Text":"Discrete variables are variables whose values come from a series of possible values."},{"Start":"04:28.630 ","End":"04:31.680","Text":"For example, the number of children in a family, well,"},{"Start":"04:31.680 ","End":"04:33.030","Text":"that could be 0, 1,"},{"Start":"04:33.030 ","End":"04:35.065","Text":"2, and 3 and so on and so forth."},{"Start":"04:35.065 ","End":"04:41.915","Text":"Or another example, the marks in an exam could be from 0-100 at intervals of 1."},{"Start":"04:41.915 ","End":"04:48.295","Text":"In these 2 examples, we see a pre-defined series of possible values and we also see,"},{"Start":"04:48.295 ","End":"04:52.480","Text":"let\u0027s say, a constant interval between the various values."},{"Start":"04:52.480 ","End":"04:56.710","Text":"But that\u0027s not necessarily the case in discrete variables."},{"Start":"04:56.710 ","End":"05:01.970","Text":"Let\u0027s take an example of a coin where 1 side of the coin has"},{"Start":"05:01.970 ","End":"05:07.675","Text":"the value minus 1 and the other side of the coin has the value 2.5."},{"Start":"05:07.675 ","End":"05:09.200","Text":"When you toss a coin,"},{"Start":"05:09.200 ","End":"05:11.210","Text":"whichever number pops up,"},{"Start":"05:11.210 ","End":"05:14.320","Text":"that\u0027s the amount of money that you win."},{"Start":"05:14.320 ","End":"05:17.269","Text":"Here, we\u0027re talking about discrete variable,"},{"Start":"05:17.269 ","End":"05:20.540","Text":"the outcome of the toss of a coin,"},{"Start":"05:20.540 ","End":"05:23.680","Text":"and we have a series of possible values,"},{"Start":"05:23.680 ","End":"05:25.945","Text":"that will be minus 1 and 2.5."},{"Start":"05:25.945 ","End":"05:29.895","Text":"The values don\u0027t have to be whole numbers,"},{"Start":"05:29.895 ","End":"05:32.765","Text":"and they also don\u0027t have to be positive whole numbers."},{"Start":"05:32.765 ","End":"05:40.880","Text":"We can have minus 1 where I lose some money or 2.5 where I win some money."},{"Start":"05:40.880 ","End":"05:43.610","Text":"But here again, we don\u0027t have whole numbers,"},{"Start":"05:43.610 ","End":"05:47.125","Text":"we don\u0027t have the numbers 1 and 2, we have 2.5."},{"Start":"05:47.125 ","End":"05:49.430","Text":"Here we don\u0027t have positive numbers,"},{"Start":"05:49.430 ","End":"05:52.280","Text":"we have a negative number, that\u0027s where we lose money."},{"Start":"05:52.280 ","End":"05:57.745","Text":"This then is what defines a discrete variable."},{"Start":"05:57.745 ","End":"06:01.610","Text":"On the other hand, continuous variables are variables whose values"},{"Start":"06:01.610 ","End":"06:05.210","Text":"come from an infinite number of values in a specific range."},{"Start":"06:05.210 ","End":"06:09.110","Text":"The values are continuous and are not grouped in intervals of values."},{"Start":"06:09.110 ","End":"06:10.660","Text":"For example, height."},{"Start":"06:10.660 ","End":"06:13.155","Text":"If, for example, the lowest height is"},{"Start":"06:13.155 ","End":"06:17.840","Text":"150 centimeters and the tallest is 190 centimeters,"},{"Start":"06:17.840 ","End":"06:19.700","Text":"well, the heights are continuous."},{"Start":"06:19.700 ","End":"06:24.080","Text":"There\u0027s an infinite number of possible values for height between say,"},{"Start":"06:24.080 ","End":"06:27.185","Text":"160 and 161 centimeters."},{"Start":"06:27.185 ","End":"06:30.920","Text":"For example, 160.233 centimeters"},{"Start":"06:30.920 ","End":"06:34.735","Text":"is also a possible height and it\u0027s a very valid height as well."},{"Start":"06:34.735 ","End":"06:37.985","Text":"Weight, that\u0027s also a continuous variable."},{"Start":"06:37.985 ","End":"06:39.500","Text":"Weight could be from,"},{"Start":"06:39.500 ","End":"06:42.050","Text":"let\u0027s say 70-90 kilograms."},{"Start":"06:42.050 ","End":"06:47.185","Text":"82.32214 is also a valid weight."},{"Start":"06:47.185 ","End":"06:51.260","Text":"Here we have an infinite number of weights between the range"},{"Start":"06:51.260 ","End":"06:55.325","Text":"that I\u0027ve given, from 70-90 kilograms."},{"Start":"06:55.325 ","End":"06:57.155","Text":"Well, the same goes for speed."},{"Start":"06:57.155 ","End":"07:03.980","Text":"Let\u0027s say if we have a range from 100-120 kilometers per hour."},{"Start":"07:03.980 ","End":"07:10.340","Text":"Well, we can have a speed of 111.2345 kilometers per hour."},{"Start":"07:10.340 ","End":"07:11.960","Text":"That\u0027s a valid speed."},{"Start":"07:11.960 ","End":"07:15.740","Text":"Here we see that we have an infinite number"},{"Start":"07:15.740 ","End":"07:22.245","Text":"of values for this variable in a specific interval."},{"Start":"07:22.245 ","End":"07:26.105","Text":"There we go. This is the definition of a continuous variable"},{"Start":"07:26.105 ","End":"07:31.530","Text":"and that\u0027s also the difference between a continuous and a discrete variable."}],"ID":13289},{"Watched":false,"Name":"Exercise 1","Duration":"3m 23s","ChapterTopicVideoID":12811,"CourseChapterTopicPlaylistID":265038,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12811.jpeg","UploadDate":"2018-07-30T17:09:20.2670000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this question, we\u0027re asked to classify the following variables as qualitative,"},{"Start":"00:03.570 ","End":"00:06.570","Text":"quantitative discrete, or quantitative continuous."},{"Start":"00:06.570 ","End":"00:10.620","Text":"The number of apartments in a building,"},{"Start":"00:10.620 ","End":"00:12.120","Text":"could be 5 apartments,"},{"Start":"00:12.120 ","End":"00:13.765","Text":"could have 10 apartments,"},{"Start":"00:13.765 ","End":"00:16.950","Text":"a building can have 30 apartments and so on and so forth."},{"Start":"00:16.950 ","End":"00:21.645","Text":"Here, we\u0027re dealing with something that\u0027s quantitative in nature."},{"Start":"00:21.645 ","End":"00:25.350","Text":"We\u0027re representing it by numbers here."},{"Start":"00:25.350 ","End":"00:28.440","Text":"There is a concept of more or less,"},{"Start":"00:28.440 ","End":"00:32.370","Text":"like 1 building has more apartments, than another building."},{"Start":"00:32.370 ","End":"00:36.270","Text":"We\u0027re dealing here with a quantitative variable."},{"Start":"00:36.270 ","End":"00:38.535","Text":"Now, is it discrete or continuous?"},{"Start":"00:38.535 ","End":"00:40.530","Text":"Well, between 2 values."},{"Start":"00:40.530 ","End":"00:44.340","Text":"Let\u0027s say a building having 5 apartments,"},{"Start":"00:44.340 ","End":"00:46.820","Text":"and a building having 6 apartments."},{"Start":"00:46.820 ","End":"00:51.110","Text":"We have an infinite number of apartments within that range,"},{"Start":"00:51.110 ","End":"00:52.250","Text":"between 5 and 6."},{"Start":"00:52.250 ","End":"00:53.795","Text":"Well, obviously not."},{"Start":"00:53.795 ","End":"00:58.070","Text":"This then, is a discrete variable."},{"Start":"00:58.070 ","End":"01:00.100","Text":"Now what about a person\u0027s age in years?"},{"Start":"01:00.100 ","End":"01:02.765","Text":"A person can be 10 years old, 12 years old,"},{"Start":"01:02.765 ","End":"01:04.745","Text":"20 years old, 21 years old,"},{"Start":"01:04.745 ","End":"01:06.155","Text":"and so on and so forth."},{"Start":"01:06.155 ","End":"01:10.820","Text":"Again, we have here a quantitative variable."},{"Start":"01:10.820 ","End":"01:13.070","Text":"We\u0027re dealing with numbers."},{"Start":"01:13.070 ","End":"01:15.170","Text":"We\u0027re dealing with the concept of more or less."},{"Start":"01:15.170 ","End":"01:20.150","Text":"A person is older than the next person beside him."},{"Start":"01:20.150 ","End":"01:24.665","Text":"Now, is it a continuous or is it a discrete variable?"},{"Start":"01:24.665 ","End":"01:30.830","Text":"Well, do we have an infinite number of values for ages in years between,"},{"Start":"01:30.830 ","End":"01:33.935","Text":"let\u0027s say, 20 years and 21 years of age?"},{"Start":"01:33.935 ","End":"01:36.510","Text":"Well, yes, we do. We can have,"},{"Start":"01:36.510 ","End":"01:39.045","Text":"let\u0027s say 21 and 1/2,"},{"Start":"01:39.045 ","End":"01:43.695","Text":"21.438 for example, years."},{"Start":"01:43.695 ","End":"01:47.795","Text":"This then is a continuous variable."},{"Start":"01:47.795 ","End":"01:50.000","Text":"Now what about the unemployment rate in a city?"},{"Start":"01:50.000 ","End":"01:52.190","Text":"Well, the unemployment rate in a city,"},{"Start":"01:52.190 ","End":"01:53.480","Text":"it could be 5 percent,"},{"Start":"01:53.480 ","End":"01:55.745","Text":"7 percent, 13 percent."},{"Start":"01:55.745 ","End":"02:00.810","Text":"Again, we\u0027re dealing here with a quantitative variable."},{"Start":"02:00.810 ","End":"02:02.700","Text":"There is a concept of more or less,"},{"Start":"02:02.700 ","End":"02:07.250","Text":"1 city has an unemployment rate which is higher than the next city."},{"Start":"02:07.250 ","End":"02:12.170","Text":"Now, is it a discrete or continuous variable?"},{"Start":"02:12.170 ","End":"02:18.200","Text":"Well, we can have an infinite number of unemployment rates between,"},{"Start":"02:18.200 ","End":"02:21.275","Text":"let\u0027s say 8 percent and 9 percent."},{"Start":"02:21.275 ","End":"02:25.705","Text":"For example, 8.346 percent."},{"Start":"02:25.705 ","End":"02:29.190","Text":"This then is a continuous variable,"},{"Start":"02:29.190 ","End":"02:31.580","Text":"so quantitative continuous variable."},{"Start":"02:31.580 ","End":"02:34.085","Text":"Now what about a preferred school subject?"},{"Start":"02:34.085 ","End":"02:36.080","Text":"Well, for each student, it\u0027s different."},{"Start":"02:36.080 ","End":"02:38.110","Text":"For 1 student could be Math,"},{"Start":"02:38.110 ","End":"02:40.845","Text":"another student could be Stats,"},{"Start":"02:40.845 ","End":"02:44.715","Text":"another student could be History."},{"Start":"02:44.715 ","End":"02:48.230","Text":"Here, we don\u0027t use numbers,"},{"Start":"02:48.230 ","End":"02:50.900","Text":"we have names, we have categories."},{"Start":"02:50.900 ","End":"02:54.940","Text":"Therefore, this is a qualitative variable."},{"Start":"02:54.940 ","End":"02:57.335","Text":"Even though we codify,"},{"Start":"02:57.335 ","End":"03:01.130","Text":"we can codify each 1 of these categories were Math equals 1,"},{"Start":"03:01.130 ","End":"03:03.710","Text":"Stats equals 2, History equals 3."},{"Start":"03:03.710 ","End":"03:08.275","Text":"Well again, there\u0027s no concept of more here, more or less."},{"Start":"03:08.275 ","End":"03:17.420","Text":"There\u0027s no meaning to the addition or subtraction of the codes of the school subject,"},{"Start":"03:17.420 ","End":"03:23.800","Text":"so this then is a qualitative variable."}],"ID":13290},{"Watched":false,"Name":"Exercise 2","Duration":"2m 27s","ChapterTopicVideoID":12812,"CourseChapterTopicPlaylistID":265038,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12812.jpeg","UploadDate":"2018-07-30T17:09:26.5430000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this question we\u0027re given the following table."},{"Start":"00:02.700 ","End":"00:05.400","Text":"This is a probability distribution of the number of"},{"Start":"00:05.400 ","End":"00:09.060","Text":"times that employees at the Star Company were late getting to work."},{"Start":"00:09.060 ","End":"00:11.534","Text":"The company has 200 employees."},{"Start":"00:11.534 ","End":"00:13.300","Text":"What do we have here in the table?"},{"Start":"00:13.300 ","End":"00:16.260","Text":"We have, let\u0027s say, 17 employees."},{"Start":"00:16.260 ","End":"00:20.295","Text":"We see here 17 employees that weren\u0027t late to work at all."},{"Start":"00:20.295 ","End":"00:24.285","Text":"23 employees out of the 200 that were late to work once,"},{"Start":"00:24.285 ","End":"00:27.390","Text":"85 employees out of the 200 that were late to work twice,"},{"Start":"00:27.390 ","End":"00:28.740","Text":"and so on and so forth."},{"Start":"00:28.740 ","End":"00:31.875","Text":"We\u0027re asked what\u0027s the variable being researched?"},{"Start":"00:31.875 ","End":"00:37.765","Text":"In theoretical statistics, we\u0027re focusing on a specific group,"},{"Start":"00:37.765 ","End":"00:40.760","Text":"and within that group we\u0027re"},{"Start":"00:40.760 ","End":"00:45.545","Text":"focusing on a specific variable and we want to analyze that variable."},{"Start":"00:45.545 ","End":"00:47.735","Text":"What\u0027s the group here?"},{"Start":"00:47.735 ","End":"00:57.135","Text":"That\u0027s the 200 employees at the Star company."},{"Start":"00:57.135 ","End":"01:00.860","Text":"What\u0027s the variable that\u0027s being researched?"},{"Start":"01:00.860 ","End":"01:02.675","Text":"That\u0027s this guy right here."},{"Start":"01:02.675 ","End":"01:06.800","Text":"That\u0027s the number of times that a person is late."},{"Start":"01:06.800 ","End":"01:08.360","Text":"How do I know that?"},{"Start":"01:08.360 ","End":"01:13.325","Text":"I have to ask the question for each member of this group what changes?"},{"Start":"01:13.325 ","End":"01:15.740","Text":"What\u0027s the variable?"},{"Start":"01:15.740 ","End":"01:18.139","Text":"What is the thing that we\u0027re studying?"},{"Start":"01:18.139 ","End":"01:22.520","Text":"We\u0027re studying the number of times that a person is late."},{"Start":"01:22.520 ","End":"01:29.850","Text":"The variable here was the number of times that an employee is late,"},{"Start":"01:29.850 ","End":"01:33.470","Text":"and we\u0027re asked then is this variable,"},{"Start":"01:33.470 ","End":"01:35.630","Text":"is it qualitative or quantitative?"},{"Start":"01:35.630 ","End":"01:38.915","Text":"If it\u0027s quantitative, is it discrete or continuous?"},{"Start":"01:38.915 ","End":"01:45.154","Text":"Here, we\u0027re obviously talking about a quantitative variable."},{"Start":"01:45.154 ","End":"01:47.930","Text":"Why is that? Because first of all,"},{"Start":"01:47.930 ","End":"01:52.030","Text":"we can represent the values of this variable in numbers."},{"Start":"01:52.030 ","End":"01:55.430","Text":"There is a concept of more or less."},{"Start":"01:55.430 ","End":"02:00.740","Text":"There\u0027s a group of people who were late more times than a different group of people."},{"Start":"02:00.740 ","End":"02:04.400","Text":"Is it discrete or is it continuous?"},{"Start":"02:04.400 ","End":"02:09.550","Text":"Obviously, you can\u0027t be late 2.5 times,"},{"Start":"02:09.550 ","End":"02:11.745","Text":"or 2 a third times."},{"Start":"02:11.745 ","End":"02:16.065","Text":"This then is a discrete variable."},{"Start":"02:16.065 ","End":"02:21.080","Text":"We have here the number of times that an employee is late,"},{"Start":"02:21.080 ","End":"02:22.310","Text":"that\u0027s a variable,"},{"Start":"02:22.310 ","End":"02:25.830","Text":"and it\u0027s a quantitative discrete variable."}],"ID":13291},{"Watched":false,"Name":"Exercise 3","Duration":"2m 12s","ChapterTopicVideoID":12813,"CourseChapterTopicPlaylistID":265038,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12813.jpeg","UploadDate":"2018-07-30T17:09:38.7430000","DurationForVideoObject":"PT2M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this question, we\u0027re given a list of quantitative variables and we\u0027re asked"},{"Start":"00:03.870 ","End":"00:08.940","Text":"whether these variables are either continuous or discrete."},{"Start":"00:08.940 ","End":"00:11.445","Text":"Well, let\u0027s take a look at the first 1; wages."},{"Start":"00:11.445 ","End":"00:14.205","Text":"Wages is a continuous variable."},{"Start":"00:14.205 ","End":"00:17.760","Text":"Now, why is that? Well, we don\u0027t get wages in the whole dollars only."},{"Start":"00:17.760 ","End":"00:19.600","Text":"We can get this in partial dollars only."},{"Start":"00:19.600 ","End":"00:24.720","Text":"It\u0027s called cents. We don\u0027t get either $10 or $11."},{"Start":"00:24.720 ","End":"00:28.005","Text":"We can have $10.23 cents."},{"Start":"00:28.005 ","End":"00:30.570","Text":"Therefore, this is a continuous variable."},{"Start":"00:30.570 ","End":"00:33.460","Text":"Now, what about marks and matriculation exam?"},{"Start":"00:33.460 ","End":"00:38.750","Text":"Well, here the marks on an exam could be from 0-100,"},{"Start":"00:38.750 ","End":"00:43.535","Text":"and the increments in grades are 1 percent."},{"Start":"00:43.535 ","End":"00:49.430","Text":"We can go from 50-51 percent or from 75-76 percent."},{"Start":"00:49.430 ","End":"00:51.200","Text":"We don\u0027t have partial grades here."},{"Start":"00:51.200 ","End":"00:55.520","Text":"This then is a discrete variable."},{"Start":"00:55.520 ","End":"00:57.650","Text":"What about results of rolling a die?"},{"Start":"00:57.650 ","End":"01:00.980","Text":"Well, here obviously this is a discrete variable."},{"Start":"01:00.980 ","End":"01:09.230","Text":"Why is that? Well, we only have 6 possible values in a set of values of a die."},{"Start":"01:09.230 ","End":"01:13.360","Text":"A die can only have 6 possible values; from 1-6."},{"Start":"01:13.360 ","End":"01:15.890","Text":"This then is a discrete variable."},{"Start":"01:15.890 ","End":"01:18.650","Text":"Now, what about the speed in a race?"},{"Start":"01:18.650 ","End":"01:22.370","Text":"Well, let\u0027s say it doesn\u0027t matter if it\u0027s in miles"},{"Start":"01:22.370 ","End":"01:25.955","Text":"per hour or kilometers per second or what have you."},{"Start":"01:25.955 ","End":"01:29.675","Text":"Let\u0027s say we\u0027re talking about kilometers per hour."},{"Start":"01:29.675 ","End":"01:35.105","Text":"The speed and the race could be 10 or 11 or 15 kilometers per hour."},{"Start":"01:35.105 ","End":"01:39.920","Text":"Now, there are an infinite number of speeds of values between,"},{"Start":"01:39.920 ","End":"01:42.500","Text":"let\u0027s say, 10 and 11 kilometers per hour."},{"Start":"01:42.500 ","End":"01:46.220","Text":"This then is a continuous variable."},{"Start":"01:46.220 ","End":"01:50.435","Text":"Now, what about the proportion of citizens supporting their government?"},{"Start":"01:50.435 ","End":"01:52.519","Text":"Well, when we\u0027re talking about proportions,"},{"Start":"01:52.519 ","End":"01:53.630","Text":"we\u0027re talking about rates,"},{"Start":"01:53.630 ","End":"01:55.520","Text":"we are talking about percentages."},{"Start":"01:55.520 ","End":"01:59.360","Text":"Well, between 2 values there\u0027s an infinite number of values."},{"Start":"01:59.360 ","End":"02:02.825","Text":"Let\u0027s say, between 30 percent and 31 percent,"},{"Start":"02:02.825 ","End":"02:07.760","Text":"there\u0027s an infinite number of percentages of citizens supporting the government."},{"Start":"02:07.760 ","End":"02:12.359","Text":"This then is a continuous variable."}],"ID":13292}],"Thumbnail":null,"ID":265038},{"Name":"Categorization of Variables and Measuring Scales","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"8m 2s","ChapterTopicVideoID":12814,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12814.jpeg","UploadDate":"2018-07-30T17:12:27.5130000","DurationForVideoObject":"PT8M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this chapter, I\u0027ll be talking about Descriptive Statistics,"},{"Start":"00:03.090 ","End":"00:07.200","Text":"specifically the Categorization of Variables and Measuring Scales."},{"Start":"00:07.200 ","End":"00:11.400","Text":"Descriptive statistics is a subject where people learn how to collect,"},{"Start":"00:11.400 ","End":"00:13.755","Text":"present, and analyze data."},{"Start":"00:13.755 ","End":"00:17.310","Text":"In descriptive statistics, we focus on a particular group."},{"Start":"00:17.310 ","End":"00:19.740","Text":"This group can be anything that we want."},{"Start":"00:19.740 ","End":"00:22.890","Text":"It could be children in a kindergarten class,"},{"Start":"00:22.890 ","End":"00:25.800","Text":"it could be stocks in a portfolio,"},{"Start":"00:25.800 ","End":"00:28.499","Text":"it could be a place of residence."},{"Start":"00:28.499 ","End":"00:33.240","Text":"We study this group by collecting data about the group\u0027s members."},{"Start":"00:33.240 ","End":"00:35.220","Text":"How do we collect data?"},{"Start":"00:35.220 ","End":"00:36.240","Text":"How do we store it?"},{"Start":"00:36.240 ","End":"00:39.705","Text":"We use a mechanism called the variable."},{"Start":"00:39.705 ","End":"00:41.985","Text":"So what\u0027s a variable?"},{"Start":"00:41.985 ","End":"00:46.865","Text":"A variable is a characteristic that can receive a number of values."},{"Start":"00:46.865 ","End":"00:50.885","Text":"For example, a political opinion, that\u0027s a variable."},{"Start":"00:50.885 ","End":"00:53.930","Text":"The data that I can store in"},{"Start":"00:53.930 ","End":"00:58.760","Text":"a political opinion or the values that a political opinion can have,"},{"Start":"00:58.760 ","End":"01:01.610","Text":"could be for example, right wing."},{"Start":"01:01.610 ","End":"01:04.205","Text":"The person can be right winged or left wing,"},{"Start":"01:04.205 ","End":"01:07.705","Text":"or he could have no political opinion whatsoever."},{"Start":"01:07.705 ","End":"01:16.190","Text":"So these are specific values that I can store in a variable called political opinion."},{"Start":"01:16.190 ","End":"01:17.975","Text":"What about place of residence?"},{"Start":"01:17.975 ","End":"01:20.690","Text":"Place of residence could be Los Angeles, New York,"},{"Start":"01:20.690 ","End":"01:23.780","Text":"Chicago, Miami, and so on and so forth."},{"Start":"01:23.780 ","End":"01:25.925","Text":"What about a height of a person?"},{"Start":"01:25.925 ","End":"01:27.625","Text":"A person can be short."},{"Start":"01:27.625 ","End":"01:30.690","Text":"Could be 155 centimeters,"},{"Start":"01:30.690 ","End":"01:31.890","Text":"or it could be average,"},{"Start":"01:31.890 ","End":"01:34.470","Text":"168 and 1/2 centimeters,"},{"Start":"01:34.470 ","End":"01:36.105","Text":"or it could be very tall."},{"Start":"01:36.105 ","End":"01:40.115","Text":"193 and 1/3 centimeter and so on and so forth."},{"Start":"01:40.115 ","End":"01:44.375","Text":"So as we can see, there are many types of variables that we have to deal with."},{"Start":"01:44.375 ","End":"01:49.280","Text":"How can we categorize these variables in order to know how to use them?"},{"Start":"01:49.280 ","End":"01:55.110","Text":"How to present them and how to analyze the data that we have."},{"Start":"01:55.120 ","End":"01:58.460","Text":"1 categorization of the measured variables is"},{"Start":"01:58.460 ","End":"02:00.980","Text":"according to what\u0027s called a measuring scale."},{"Start":"02:00.980 ","End":"02:05.420","Text":"Think of a measuring scale as a ladder with 4 rungs on it."},{"Start":"02:05.420 ","End":"02:06.695","Text":"With the bottom rung,"},{"Start":"02:06.695 ","End":"02:09.730","Text":"is the nominal scale or variables."},{"Start":"02:09.730 ","End":"02:14.160","Text":"Variables belonging to this scale right here, a nominal variable."},{"Start":"02:14.160 ","End":"02:18.800","Text":"That\u0027s a variable whose values have meaning only for identification purposes only."},{"Start":"02:18.800 ","End":"02:21.635","Text":"There is no concept of more or less here."},{"Start":"02:21.635 ","End":"02:23.635","Text":"Let\u0027s take an example,"},{"Start":"02:23.635 ","End":"02:26.460","Text":"let\u0027s take the variable, family status."},{"Start":"02:26.460 ","End":"02:29.540","Text":"The values of the family status could be single,"},{"Start":"02:29.540 ","End":"02:32.270","Text":"married, widow, or divorced, etc."},{"Start":"02:32.270 ","End":"02:37.355","Text":"There\u0027s no concept of more or less in this variable."},{"Start":"02:37.355 ","End":"02:41.000","Text":"1 can\u0027t be more married than another person is single."},{"Start":"02:41.000 ","End":"02:45.575","Text":"On the other hand, you can\u0027t multiply being a window by 3."},{"Start":"02:45.575 ","End":"02:47.800","Text":"Just doesn\u0027t make any sense."},{"Start":"02:47.800 ","End":"02:50.060","Text":"Let\u0027s take a look at another example."},{"Start":"02:50.060 ","End":"02:53.695","Text":"The variable called place of residence."},{"Start":"02:53.695 ","End":"02:57.020","Text":"Here the valid values could be Los Angeles,"},{"Start":"02:57.020 ","End":"03:00.785","Text":"Chicago, New York, Miami, or what have you."},{"Start":"03:00.785 ","End":"03:03.800","Text":"Again, there\u0027s no concept of more or less here."},{"Start":"03:03.800 ","End":"03:07.955","Text":"You can\u0027t be living more in Miami than you do in Chicago."},{"Start":"03:07.955 ","End":"03:11.525","Text":"You can\u0027t multiply living in New York by 3."},{"Start":"03:11.525 ","End":"03:14.405","Text":"Again, it just doesn\u0027t make sense."},{"Start":"03:14.405 ","End":"03:19.940","Text":"So these are variables that belong to the nominal scale right here."},{"Start":"03:19.940 ","End":"03:24.770","Text":"There\u0027s a special case here of a variable belonging to the nominal scale."},{"Start":"03:24.770 ","End":"03:27.320","Text":"It\u0027s called a dichotomous variable."},{"Start":"03:27.320 ","End":"03:32.430","Text":"These variables are variables with only 2 possible values."},{"Start":"03:32.430 ","End":"03:36.750","Text":"For example, gender: male and female."},{"Start":"03:36.750 ","End":"03:39.580","Text":"The variable would be gender,"},{"Start":"03:39.580 ","End":"03:45.515","Text":"and the valid values would be either male or female, smoker or non-smoker."},{"Start":"03:45.515 ","End":"03:49.055","Text":"That\u0027s another example of a dichotomous variable."},{"Start":"03:49.055 ","End":"03:52.475","Text":"What about tossing a coin?"},{"Start":"03:52.475 ","End":"03:54.185","Text":"We have heads or tails."},{"Start":"03:54.185 ","End":"03:56.635","Text":"So that\u0027s the third example."},{"Start":"03:56.635 ","End":"04:00.215","Text":"The next rung on the ladder is the ordinal scale."},{"Start":"04:00.215 ","End":"04:03.080","Text":"Variables belonging to the ordinal scale are"},{"Start":"04:03.080 ","End":"04:07.055","Text":"variables whose value in addition to nominal characteristics,"},{"Start":"04:07.055 ","End":"04:10.820","Text":"have order as well but the difference between the values has no meaning."},{"Start":"04:10.820 ","End":"04:13.175","Text":"For example, military rank."},{"Start":"04:13.175 ","End":"04:14.690","Text":"We have privates,"},{"Start":"04:14.690 ","End":"04:17.630","Text":"and then we have corporals and then we have sergeants."},{"Start":"04:17.630 ","End":"04:25.415","Text":"So apart from the fact that each rank has a name that\u0027s the nominal characteristic,"},{"Start":"04:25.415 ","End":"04:28.730","Text":"there\u0027s also meaning to the order of these ranks."},{"Start":"04:28.730 ","End":"04:31.220","Text":"A corporal is more than a private,"},{"Start":"04:31.220 ","End":"04:33.200","Text":"and a sergeant is more than a corporal."},{"Start":"04:33.200 ","End":"04:35.430","Text":"But on the other hand,"},{"Start":"04:35.480 ","End":"04:41.015","Text":"2 sergeants minus 3 corporals doesn\u0027t have any meaning whatsoever."},{"Start":"04:41.015 ","End":"04:46.520","Text":"There\u0027s no meaning to the difference between the values in the ordinal scale."},{"Start":"04:46.520 ","End":"04:52.255","Text":"So this then is the difference between ordinal and nominal scales."},{"Start":"04:52.255 ","End":"04:56.160","Text":"The next rung on the ladder is the Interval scale."},{"Start":"04:56.160 ","End":"05:01.055","Text":"Variables belonging to the interval scale are variables whose values are ordered."},{"Start":"05:01.055 ","End":"05:03.470","Text":"The difference between the values has meaning,"},{"Start":"05:03.470 ","End":"05:06.290","Text":"but the ratio between the values has no meaning."},{"Start":"05:06.290 ","End":"05:10.640","Text":"This category isn\u0027t very common so we won\u0027t be getting into that much."},{"Start":"05:10.640 ","End":"05:12.020","Text":"But let\u0027s give an example."},{"Start":"05:12.020 ","End":"05:14.825","Text":"The example being, the number of floors in a building."},{"Start":"05:14.825 ","End":"05:17.860","Text":"Let\u0027s take a building with 4 floors,"},{"Start":"05:17.860 ","End":"05:19.755","Text":"while each floor has a name."},{"Start":"05:19.755 ","End":"05:22.185","Text":"Floor number 1, floor number 2."},{"Start":"05:22.185 ","End":"05:26.850","Text":"Floor number 1 is the first floor\u0027s name and floor number 4,"},{"Start":"05:26.850 ","End":"05:29.475","Text":"that\u0027s the fourth floor name."},{"Start":"05:29.475 ","End":"05:32.765","Text":"So we have nominal characteristics."},{"Start":"05:32.765 ","End":"05:35.645","Text":"There is meaning to the order here."},{"Start":"05:35.645 ","End":"05:40.940","Text":"The meaning is that floor number 4 is higher than floor number 3,"},{"Start":"05:40.940 ","End":"05:42.650","Text":"which is higher than floor number 2,"},{"Start":"05:42.650 ","End":"05:44.000","Text":"and so on and so forth."},{"Start":"05:44.000 ","End":"05:47.560","Text":"So the meaning of order here would be height."},{"Start":"05:47.560 ","End":"05:49.590","Text":"What about the difference?"},{"Start":"05:49.590 ","End":"05:53.120","Text":"The difference between the values also has meaning."},{"Start":"05:53.120 ","End":"05:58.255","Text":"For example, if we subtract the number of floor number 4,"},{"Start":"05:58.255 ","End":"06:00.919","Text":"which is 4 minus 3,"},{"Start":"06:00.919 ","End":"06:03.320","Text":"the number of flow number 3,"},{"Start":"06:03.320 ","End":"06:09.350","Text":"this difference means that there\u0027s a difference of 1 floor between floor numbers 3 and 4."},{"Start":"06:09.350 ","End":"06:11.875","Text":"So that\u0027s the meaning of the difference."},{"Start":"06:11.875 ","End":"06:15.410","Text":"But, the ratio between the values has no meaning."},{"Start":"06:15.410 ","End":"06:18.785","Text":"If we divide floor number 3 by flow number 2,"},{"Start":"06:18.785 ","End":"06:20.270","Text":"there\u0027s no meaning into it,"},{"Start":"06:20.270 ","End":"06:22.730","Text":"there\u0027s that just doesn\u0027t make sense."},{"Start":"06:22.730 ","End":"06:27.340","Text":"So this then is the interval scale."},{"Start":"06:27.350 ","End":"06:31.800","Text":"The last rung in the ladder is the ratio scale."},{"Start":"06:31.800 ","End":"06:36.485","Text":"Variables belonging to the ratio scale are variables whose values are ordered."},{"Start":"06:36.485 ","End":"06:38.870","Text":"The difference between the values has meaning and"},{"Start":"06:38.870 ","End":"06:41.585","Text":"the ratio between the values also has meaning."},{"Start":"06:41.585 ","End":"06:46.940","Text":"For example, the number of cars that a family owns or a person\u0027s weight in kilograms."},{"Start":"06:46.940 ","End":"06:52.250","Text":"The easiest way to identify the scale as a ratio scale is through the 0 test."},{"Start":"06:52.250 ","End":"06:53.960","Text":"In a ratio scale,"},{"Start":"06:53.960 ","End":"06:58.100","Text":"0 is the absolute representing the absence of something."},{"Start":"06:58.100 ","End":"07:00.500","Text":"Let\u0027s take this example right here."},{"Start":"07:00.500 ","End":"07:03.920","Text":"If the number of cars that a family owns equals to 0,"},{"Start":"07:03.920 ","End":"07:06.365","Text":"then the family doesn\u0027t own any cars."},{"Start":"07:06.365 ","End":"07:08.540","Text":"There\u0027s an absence of cars here,"},{"Start":"07:08.540 ","End":"07:09.995","Text":"an absence of something."},{"Start":"07:09.995 ","End":"07:15.440","Text":"So that means that the number of cars that a family has passes the 0 test therefore,"},{"Start":"07:15.440 ","End":"07:17.435","Text":"it belongs to the ratio scale."},{"Start":"07:17.435 ","End":"07:22.435","Text":"On the other hand, let\u0027s take an example from the interval scale."},{"Start":"07:22.435 ","End":"07:23.890","Text":"And here\u0027s the example,"},{"Start":"07:23.890 ","End":"07:26.140","Text":"the number of floors in a building."},{"Start":"07:26.140 ","End":"07:28.525","Text":"Can that pass the 0 test?"},{"Start":"07:28.525 ","End":"07:33.155","Text":"The entrance to a building could be called floor number 0."},{"Start":"07:33.155 ","End":"07:38.410","Text":"So, does that mean that there\u0027s an absence of the entrance of the building?"},{"Start":"07:38.410 ","End":"07:43.075","Text":"Obviously not. That\u0027s just the name of the entrance, floor number 0."},{"Start":"07:43.075 ","End":"07:46.435","Text":"So that means that this variable right here,"},{"Start":"07:46.435 ","End":"07:50.425","Text":"belongs to the interval and not the ratio scale."},{"Start":"07:50.425 ","End":"07:54.400","Text":"All these rungs in the ladder, all these scales,"},{"Start":"07:54.400 ","End":"07:56.260","Text":"it may seem confusing,"},{"Start":"07:56.260 ","End":"08:02.330","Text":"so let\u0027s do exercise number 1 in order to let all this information sink in."}],"ID":13293},{"Watched":false,"Name":"Tutorial 2","Duration":"7m 15s","ChapterTopicVideoID":12819,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12819.jpeg","UploadDate":"2018-07-30T17:22:05.8270000","DurationForVideoObject":"PT7M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"Up till now, we\u0027ve talked about measuring scales."},{"Start":"00:03.045 ","End":"00:05.550","Text":"We had 4 ranks in the ladder."},{"Start":"00:05.550 ","End":"00:07.334","Text":"We had nominal variables,"},{"Start":"00:07.334 ","End":"00:10.785","Text":"ordinal variables, interval variables, and ratio variables."},{"Start":"00:10.785 ","End":"00:14.160","Text":"Another type of categorization is"},{"Start":"00:14.160 ","End":"00:18.180","Text":"variables that are qualitative or quantitative in nature."},{"Start":"00:18.180 ","End":"00:22.860","Text":"If they\u0027re quantitative, then they can be either discrete or continuous."},{"Start":"00:22.860 ","End":"00:25.740","Text":"Let\u0027s just define these categories of"},{"Start":"00:25.740 ","End":"00:29.605","Text":"variables and we\u0027ll start with qualitative and quantitative."},{"Start":"00:29.605 ","End":"00:34.740","Text":"Qualitative variables are variables whose values have no meaning of more or less."},{"Start":"00:34.740 ","End":"00:39.649","Text":"The values of these variables have no quantitative content whatsoever."},{"Start":"00:39.649 ","End":"00:42.253","Text":"For example, a person\u0027s place of residence."},{"Start":"00:42.253 ","End":"00:44.990","Text":"It could be New York, Chicago, Los Angeles."},{"Start":"00:44.990 ","End":"00:48.560","Text":"There\u0027s no meaning of more or less here."},{"Start":"00:48.560 ","End":"00:52.400","Text":"Is New York more than Chicago or Los Angeles?"},{"Start":"00:52.400 ","End":"00:56.825","Text":"We don\u0027t know that. It doesn\u0027t make any sense even to ask this question."},{"Start":"00:56.825 ","End":"01:01.235","Text":"A person\u0027s gender is another qualitative variable."},{"Start":"01:01.235 ","End":"01:04.205","Text":"A person could be either a male or a female."},{"Start":"01:04.205 ","End":"01:07.550","Text":"Is there any quantitative content here?"},{"Start":"01:07.550 ","End":"01:12.530","Text":"Do we have any meanings of more or less here? Well, we don\u0027t."},{"Start":"01:12.530 ","End":"01:15.680","Text":"Another example would be a person\u0027s family status."},{"Start":"01:15.680 ","End":"01:17.030","Text":"It could be single, married,"},{"Start":"01:17.030 ","End":"01:18.710","Text":"divorced, or a widower."},{"Start":"01:18.710 ","End":"01:21.300","Text":"Again, there\u0027s no concept here,"},{"Start":"01:21.300 ","End":"01:24.645","Text":"there is no meaning of more or less."},{"Start":"01:24.645 ","End":"01:28.160","Text":"Can a person be more married than another person is single?"},{"Start":"01:28.160 ","End":"01:33.650","Text":"Or can a person be more male than another person be female?"},{"Start":"01:33.650 ","End":"01:36.960","Text":"These things just don\u0027t make sense."},{"Start":"01:36.960 ","End":"01:43.130","Text":"As such, these variables are more of a category variable."},{"Start":"01:43.130 ","End":"01:49.500","Text":"They provide categories for the various members of our group."},{"Start":"01:49.750 ","End":"01:52.630","Text":"Let\u0027s take a look at quantitative variables."},{"Start":"01:52.630 ","End":"01:56.960","Text":"These are variables whose values are numbers with quantitative meaning,"},{"Start":"01:56.960 ","End":"01:58.865","Text":"for example, a person\u0027s height."},{"Start":"01:58.865 ","End":"02:02.620","Text":"We can have a tall person or a short person."},{"Start":"02:02.620 ","End":"02:04.145","Text":"We\u0027re using numbers."},{"Start":"02:04.145 ","End":"02:09.050","Text":"The number of centimeters to depict the height of a person."},{"Start":"02:09.050 ","End":"02:12.560","Text":"The height of a person has quantitative meanings."},{"Start":"02:12.560 ","End":"02:13.760","Text":"The higher the number,"},{"Start":"02:13.760 ","End":"02:15.140","Text":"the taller the person."},{"Start":"02:15.140 ","End":"02:17.150","Text":"That\u0027s a quantitative meaning."},{"Start":"02:17.150 ","End":"02:20.020","Text":"Another example would be marks in an exam."},{"Start":"02:20.020 ","End":"02:24.495","Text":"We\u0027re using numbers to depict the mark on an exam in percentages."},{"Start":"02:24.495 ","End":"02:26.480","Text":"It could be 99 percent,"},{"Start":"02:26.480 ","End":"02:28.250","Text":"it could be 65 percent."},{"Start":"02:28.250 ","End":"02:30.850","Text":"The higher the mark,"},{"Start":"02:30.850 ","End":"02:36.435","Text":"that means that the student excelled more on the exam."},{"Start":"02:36.435 ","End":"02:39.574","Text":"That\u0027s the meaning of quantitative variables."},{"Start":"02:39.574 ","End":"02:42.605","Text":"There are 2 types of quantitative variables,"},{"Start":"02:42.605 ","End":"02:46.215","Text":"discrete and continuous variables."},{"Start":"02:46.215 ","End":"02:49.160","Text":"Let\u0027s take a look at discrete variables."},{"Start":"02:49.160 ","End":"02:53.240","Text":"These are variables whose values come from a series of possible values."},{"Start":"02:53.240 ","End":"02:57.320","Text":"This series has a finite number of possible values."},{"Start":"02:57.320 ","End":"02:59.210","Text":"Let\u0027s take a look at an example."},{"Start":"02:59.210 ","End":"03:01.100","Text":"The number of children in a family."},{"Start":"03:01.100 ","End":"03:06.140","Text":"A family can have 0 or 1 or 2 children if it\u0027s a small family,"},{"Start":"03:06.140 ","End":"03:07.700","Text":"or if it\u0027s a larger family,"},{"Start":"03:07.700 ","End":"03:09.485","Text":"if we take it even to the extreme,"},{"Start":"03:09.485 ","End":"03:11.960","Text":"they can have 15 or 20 children."},{"Start":"03:11.960 ","End":"03:15.845","Text":"But they certainly can\u0027t have 100 children or 200 children."},{"Start":"03:15.845 ","End":"03:18.890","Text":"What about marks on an exam?"},{"Start":"03:18.890 ","End":"03:23.345","Text":"Well, the marks on an exam can be from either 0 on 1 hand,"},{"Start":"03:23.345 ","End":"03:28.370","Text":"and on the other, we can excel it and get 100 percent."},{"Start":"03:28.370 ","End":"03:31.765","Text":"These marks are given to us at intervals of 1."},{"Start":"03:31.765 ","End":"03:34.280","Text":"Again, in both these examples,"},{"Start":"03:34.280 ","End":"03:38.210","Text":"we see that we have some finite set"},{"Start":"03:38.210 ","End":"03:43.055","Text":"of values that a discrete variable can have but no more."},{"Start":"03:43.055 ","End":"03:46.160","Text":"Let\u0027s take a look at continuous variables."},{"Start":"03:46.160 ","End":"03:48.740","Text":"These are variables whose values come from"},{"Start":"03:48.740 ","End":"03:51.785","Text":"an infinite number of values in a specific range."},{"Start":"03:51.785 ","End":"03:56.150","Text":"The values are continuous and aren\u0027t grouped in intervals of values."},{"Start":"03:56.150 ","End":"03:59.270","Text":"Let\u0027s just take a look at some examples."},{"Start":"03:59.270 ","End":"04:02.495","Text":"Let\u0027s take a look at height, for example."},{"Start":"04:02.495 ","End":"04:06.215","Text":"If for example, the lowest height is 150 centimeters,"},{"Start":"04:06.215 ","End":"04:09.560","Text":"and the tallest is 190 centimeters,"},{"Start":"04:09.560 ","End":"04:11.269","Text":"the heights are continuous."},{"Start":"04:11.269 ","End":"04:13.445","Text":"We have a range here."},{"Start":"04:13.445 ","End":"04:20.105","Text":"There\u0027s an infinite number of possible values for height between 160 and 161 centimeters."},{"Start":"04:20.105 ","End":"04:24.665","Text":"We could have a valid value of 160.233 centimeters."},{"Start":"04:24.665 ","End":"04:26.690","Text":"That\u0027s a valid value for height."},{"Start":"04:26.690 ","End":"04:31.550","Text":"We have here, an infinite number of values that height can have"},{"Start":"04:31.550 ","End":"04:37.260","Text":"between even this range right here from 160-161 centimeters."},{"Start":"04:37.260 ","End":"04:39.555","Text":"For weight, that will be the same thing."},{"Start":"04:39.555 ","End":"04:41.870","Text":"If we have a range between,"},{"Start":"04:41.870 ","End":"04:45.125","Text":"let\u0027s say, 75 and 76 kilograms."},{"Start":"04:45.125 ","End":"04:51.275","Text":"Well, there\u0027s an infinite number of values for weight in this 1 kilogram range."},{"Start":"04:51.275 ","End":"04:52.640","Text":"Same thing for speed."},{"Start":"04:52.640 ","End":"04:56.330","Text":"If I\u0027m driving a car at 100 kilometers per hour,"},{"Start":"04:56.330 ","End":"05:01.595","Text":"and let\u0027s say I move it up to 101 kilometers per hour."},{"Start":"05:01.595 ","End":"05:09.064","Text":"Well, I just passed an infinite number of values in that 1 kilometer per hour range,"},{"Start":"05:09.064 ","End":"05:10.505","Text":"and so on and so forth."},{"Start":"05:10.505 ","End":"05:12.815","Text":"I hope you get the idea here."},{"Start":"05:12.815 ","End":"05:20.575","Text":"This then is the definition of continuous variables. Let\u0027s sum things up."},{"Start":"05:20.575 ","End":"05:24.265","Text":"In theoretical statistics, we\u0027re interested in groups."},{"Start":"05:24.265 ","End":"05:27.295","Text":"We\u0027re interested in collecting data from groups,"},{"Start":"05:27.295 ","End":"05:29.200","Text":"we\u0027re interested in presenting the data,"},{"Start":"05:29.200 ","End":"05:30.670","Text":"and analyzing the data."},{"Start":"05:30.670 ","End":"05:32.350","Text":"What\u0027s a group?"},{"Start":"05:32.350 ","End":"05:34.720","Text":"A group could be basically anything."},{"Start":"05:34.720 ","End":"05:36.519","Text":"It could be a group of stocks,"},{"Start":"05:36.519 ","End":"05:38.965","Text":"it could be a group of buildings,"},{"Start":"05:38.965 ","End":"05:40.780","Text":"a group of people."},{"Start":"05:40.780 ","End":"05:45.400","Text":"For example, it could be a group of employees in a specific company."},{"Start":"05:45.400 ","End":"05:47.680","Text":"How do we collect the data,"},{"Start":"05:47.680 ","End":"05:48.953","Text":"how do we present the data,"},{"Start":"05:48.953 ","End":"05:52.450","Text":"and how do we analyze the data on this group?"},{"Start":"05:52.450 ","End":"05:55.300","Text":"Well, we have to look at the various members of this group,"},{"Start":"05:55.300 ","End":"05:59.965","Text":"and in our case, that\u0027ll be the employees in the specific company."},{"Start":"05:59.965 ","End":"06:03.350","Text":"How do we gather data about the employees?"},{"Start":"06:03.350 ","End":"06:04.625","Text":"Well, we\u0027re looking at"},{"Start":"06:04.625 ","End":"06:10.145","Text":"a specific characteristic that varies from individual to individual."},{"Start":"06:10.145 ","End":"06:12.215","Text":"This specific characteristic,"},{"Start":"06:12.215 ","End":"06:14.300","Text":"we\u0027ll call a variable."},{"Start":"06:14.300 ","End":"06:20.735","Text":"These variables can be qualitative in nature or quantitative in nature."},{"Start":"06:20.735 ","End":"06:23.270","Text":"Why do we categorize them as such?"},{"Start":"06:23.270 ","End":"06:26.600","Text":"Well, we categorize them like this because they\u0027ll give us"},{"Start":"06:26.600 ","End":"06:32.115","Text":"a better understanding of how to present the data in the variables,"},{"Start":"06:32.115 ","End":"06:34.545","Text":"and how to analyze the data."},{"Start":"06:34.545 ","End":"06:37.190","Text":"If we have qualitative variables,"},{"Start":"06:37.190 ","End":"06:41.105","Text":"for example, an example of that would be gender."},{"Start":"06:41.105 ","End":"06:44.320","Text":"An employee could be either a male or female."},{"Start":"06:44.320 ","End":"06:47.300","Text":"If we\u0027re looking at quantitative variables,"},{"Start":"06:47.300 ","End":"06:49.835","Text":"they can be either discrete or continuous."},{"Start":"06:49.835 ","End":"06:53.780","Text":"A continuous variable could be,"},{"Start":"06:53.780 ","End":"06:57.995","Text":"let\u0027s say, the number of years that a person has been working at the company."},{"Start":"06:57.995 ","End":"07:03.440","Text":"A discrete variable could be the number of computers that a person has on his desk."},{"Start":"07:03.440 ","End":"07:09.695","Text":"This then is our main focus in the next several chapters."},{"Start":"07:09.695 ","End":"07:15.210","Text":"Let\u0027s just start exercising to let everything sink in."}],"ID":13294},{"Watched":false,"Name":"Exercise 1 - Parts a-e","Duration":"6m 52s","ChapterTopicVideoID":12815,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12815.jpeg","UploadDate":"2018-07-30T17:12:48.7900000","DurationForVideoObject":"PT6M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"The following is a list of variables."},{"Start":"00:02.295 ","End":"00:04.245","Text":"Now, for each variable below,"},{"Start":"00:04.245 ","End":"00:06.060","Text":"we have to state the measuring scale,"},{"Start":"00:06.060 ","End":"00:08.339","Text":"whether it\u0027s nominal, ordinal,"},{"Start":"00:08.339 ","End":"00:10.020","Text":"interval, or ratio."},{"Start":"00:10.020 ","End":"00:13.785","Text":"Now, the first variable is the height of a person in centimeters."},{"Start":"00:13.785 ","End":"00:16.110","Text":"Now, a height of a person could be, let\u0027s say,"},{"Start":"00:16.110 ","End":"00:20.925","Text":"150 centimeters, 160, 173 centimeters."},{"Start":"00:20.925 ","End":"00:24.165","Text":"It really doesn\u0027t matter. It varies from person to person."},{"Start":"00:24.165 ","End":"00:29.130","Text":"The measuring scale for the height of a person in centimeters,"},{"Start":"00:29.130 ","End":"00:33.160","Text":"well, that\u0027s a ratio scale."},{"Start":"00:33.230 ","End":"00:38.919","Text":"That\u0027s right here. Now, the best way to identify whether a variable"},{"Start":"00:38.919 ","End":"00:44.430","Text":"has or belongs to a ratio scale or not is through the zero test."},{"Start":"00:44.430 ","End":"00:49.920","Text":"If 0 represents nothing as it does here."},{"Start":"00:49.920 ","End":"00:52.310","Text":"If we\u0027d have no height whatsoever,"},{"Start":"00:52.310 ","End":"00:56.294","Text":"if the height of a person is 0 or the height of anything is 0,"},{"Start":"00:56.294 ","End":"00:59.390","Text":"that means that there\u0027s no height and as such,"},{"Start":"00:59.390 ","End":"01:07.200","Text":"then this belongs to the ratio scale because this variable passed the zero test."},{"Start":"01:07.360 ","End":"01:10.880","Text":"The next variable is the number of children in a family."},{"Start":"01:10.880 ","End":"01:13.400","Text":"Well, a family can have 0 children,"},{"Start":"01:13.400 ","End":"01:15.860","Text":"1 child, 2 children or more."},{"Start":"01:15.860 ","End":"01:21.560","Text":"Now, it\u0027s easy to see that this variable belongs to the ratio scale."},{"Start":"01:21.560 ","End":"01:25.340","Text":"Now why is that? Well, let\u0027s take a look at the 0 test."},{"Start":"01:25.340 ","End":"01:26.990","Text":"What does 0 mean here?"},{"Start":"01:26.990 ","End":"01:31.760","Text":"Well, if a family has 0 children then they have no children whatsoever."},{"Start":"01:31.760 ","End":"01:34.670","Text":"There\u0027s an absence of children and as such,"},{"Start":"01:34.670 ","End":"01:43.330","Text":"this variable passed the zero test and therefore it belongs to the ratio scale."},{"Start":"01:43.330 ","End":"01:47.630","Text":"The next variable would be the degree of anxiety before an exam."},{"Start":"01:47.630 ","End":"01:50.210","Text":"Now, how do we measure degree of anxiety?"},{"Start":"01:50.210 ","End":"01:53.750","Text":"We don\u0027t have any technology to do that and we can\u0027t just plug in a person to"},{"Start":"01:53.750 ","End":"01:57.550","Text":"a machine in order to calculate their degree of anxiety."},{"Start":"01:57.550 ","End":"02:03.295","Text":"What we usually do is we provide that person with a scale."},{"Start":"02:03.295 ","End":"02:04.870","Text":"Now, this scale could be,"},{"Start":"02:04.870 ","End":"02:06.490","Text":"let\u0027s say a scale from 1-5."},{"Start":"02:06.490 ","End":"02:09.295","Text":"It could be from 0-10 for example."},{"Start":"02:09.295 ","End":"02:11.845","Text":"Let\u0027s say we have a scale from 0,"},{"Start":"02:11.845 ","End":"02:15.635","Text":"1, 2, and so on and so forth until 10."},{"Start":"02:15.635 ","End":"02:21.940","Text":"That would be my scale where 0 would be the least amount of anxiety and 10 would be"},{"Start":"02:21.940 ","End":"02:28.810","Text":"the highest level of anxiety and we ask a person to pick 1 of these values."},{"Start":"02:28.810 ","End":"02:33.040","Text":"Here we have let\u0027s say nominal characteristics here."},{"Start":"02:33.040 ","End":"02:37.060","Text":"0 could be the lowest level of anxiety."},{"Start":"02:37.060 ","End":"02:40.660","Text":"That\u0027s low, and that\u0027s high."},{"Start":"02:41.030 ","End":"02:43.760","Text":"Instead of low, medium,"},{"Start":"02:43.760 ","End":"02:45.920","Text":"and high, we provide numbers."},{"Start":"02:45.920 ","End":"02:51.095","Text":"We have names for the various levels of anxiety."},{"Start":"02:51.095 ","End":"02:54.380","Text":"Now, we also have ordinal characteristics here,"},{"Start":"02:54.380 ","End":"02:59.870","Text":"we know that anxiety number 3 is higher than anxiety number 1."},{"Start":"02:59.870 ","End":"03:02.960","Text":"Now, do we have an interval characteristic here?"},{"Start":"03:02.960 ","End":"03:08.690","Text":"Well, if we subtract anxiety number 7 from anxiety number 6,"},{"Start":"03:08.690 ","End":"03:10.235","Text":"is there any meaning to that?"},{"Start":"03:10.235 ","End":"03:13.550","Text":"Well, there isn\u0027t and what about the ratio test?"},{"Start":"03:13.550 ","End":"03:15.170","Text":"The zero test?"},{"Start":"03:15.170 ","End":"03:17.450","Text":"Well, let\u0027s take a look at here."},{"Start":"03:17.450 ","End":"03:21.935","Text":"0 doesn\u0027t mean an absence of something or the absence of anxiety."},{"Start":"03:21.935 ","End":"03:26.135","Text":"All it means is that this is the lowest anxiety that we have here."},{"Start":"03:26.135 ","End":"03:31.370","Text":"I could have easily reversed the scale and said that 10 would be"},{"Start":"03:31.370 ","End":"03:37.460","Text":"the lowest degree of anxiety and 0 would be the highest degree of anxiety."},{"Start":"03:37.460 ","End":"03:40.730","Text":"This thing doesn\u0027t pass the zero test."},{"Start":"03:40.730 ","End":"03:45.445","Text":"Therefore, this variable is an ordinal variable."},{"Start":"03:45.445 ","End":"03:50.930","Text":"The next variable is the satisfaction with customer service on a scale of 1-7 where"},{"Start":"03:50.930 ","End":"03:57.229","Text":"1 means completely dissatisfied and 7 means completely satisfied."},{"Start":"03:57.229 ","End":"04:01.320","Text":"Now, here we have an ordinal scale."},{"Start":"04:01.320 ","End":"04:07.040","Text":"Now, why is that? Well, I could have used words instead of numbers."},{"Start":"04:07.040 ","End":"04:09.920","Text":"The numbers just represent the level of"},{"Start":"04:09.920 ","End":"04:14.840","Text":"satisfaction and that\u0027s all and there is order to this obviously."},{"Start":"04:14.840 ","End":"04:19.695","Text":"More satisfied means that I go up the scale here."},{"Start":"04:19.695 ","End":"04:24.845","Text":"5 means that I\u0027m more satisfied than if I provided a value of 3."},{"Start":"04:24.845 ","End":"04:30.080","Text":"Now, the interval here mean there\u0027s no meaning to the subtraction."},{"Start":"04:30.080 ","End":"04:32.030","Text":"Let\u0027s say the difference between,"},{"Start":"04:32.030 ","End":"04:36.660","Text":"let\u0027s say value 6 and value 5 on"},{"Start":"04:36.660 ","End":"04:39.440","Text":"this scale and obviously this"},{"Start":"04:39.440 ","End":"04:42.740","Text":"isn\u0027t a ratio variable because it doesn\u0027t even pass the zero test."},{"Start":"04:42.740 ","End":"04:46.475","Text":"There isn\u0027t any 0 here at all."},{"Start":"04:46.475 ","End":"04:51.995","Text":"0 doesn\u0027t mean that there\u0027s an absence of satisfaction whatsoever."},{"Start":"04:51.995 ","End":"04:55.325","Text":"Therefore this is an ordinal variable."},{"Start":"04:55.325 ","End":"04:57.950","Text":"The next variable is education."},{"Start":"04:57.950 ","End":"05:01.250","Text":"Now, here we have a problem. Now, why is that?"},{"Start":"05:01.250 ","End":"05:05.615","Text":"Because we\u0027re not given the method of how education is measured."},{"Start":"05:05.615 ","End":"05:09.259","Text":"Now, if education was measured in years,"},{"Start":"05:09.259 ","End":"05:13.340","Text":"a person studied for 8 years or 10 years or 15,"},{"Start":"05:13.340 ","End":"05:18.200","Text":"20 years, then this would obviously be a ratio."},{"Start":"05:18.200 ","End":"05:23.990","Text":"Now, why is that? Well, it\u0027s clear that if somebody had 16 years of education, well,"},{"Start":"05:23.990 ","End":"05:27.385","Text":"that\u0027s twice as much education as a person who had"},{"Start":"05:27.385 ","End":"05:31.580","Text":"8 years of education and it does pass the zero test."},{"Start":"05:31.580 ","End":"05:34.820","Text":"If somebody had 0 years of education,"},{"Start":"05:34.820 ","End":"05:37.855","Text":"then that means that there\u0027s an absence of education."},{"Start":"05:37.855 ","End":"05:40.625","Text":"That person didn\u0027t study at all."},{"Start":"05:40.625 ","End":"05:45.005","Text":"Now, we can also measure education like this."},{"Start":"05:45.005 ","End":"05:50.915","Text":"We can say that somebody had a public school level of education,"},{"Start":"05:50.915 ","End":"05:53.735","Text":"or a high school level,"},{"Start":"05:53.735 ","End":"05:57.545","Text":"or a university level."},{"Start":"05:57.545 ","End":"06:00.875","Text":"Now, if we measure education in this way,"},{"Start":"06:00.875 ","End":"06:06.620","Text":"then we obviously have an ordinal variable and again, why is that?"},{"Start":"06:06.620 ","End":"06:09.050","Text":"Well, we have nominal characteristics,"},{"Start":"06:09.050 ","End":"06:11.570","Text":"we have names for each 1 of the levels;"},{"Start":"06:11.570 ","End":"06:13.640","Text":"public school, high school, and university."},{"Start":"06:13.640 ","End":"06:18.740","Text":"There is an order to it because high school is greater than public school"},{"Start":"06:18.740 ","End":"06:23.900","Text":"with respect to education but there\u0027s no meaning to interval,"},{"Start":"06:23.900 ","End":"06:27.215","Text":"there\u0027s no meaning to the difference between these things."},{"Start":"06:27.215 ","End":"06:32.240","Text":"High-school minus public school doesn\u0027t have any meaning and obviously this doesn\u0027t"},{"Start":"06:32.240 ","End":"06:37.490","Text":"pass the zero test because there is no 0 here. There we go."},{"Start":"06:37.490 ","End":"06:43.210","Text":"We need to know how education is measured to know whether we have a ratio."},{"Start":"06:43.210 ","End":"06:47.720","Text":"Whether education measured in years or is it an ordinal variable,"},{"Start":"06:47.720 ","End":"06:52.650","Text":"whether education is measured by these levels here."}],"ID":13295},{"Watched":false,"Name":"Exercise 1 - Parts f-i","Duration":"4m 6s","ChapterTopicVideoID":12816,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12816.jpeg","UploadDate":"2018-07-30T17:13:08.7770000","DurationForVideoObject":"PT4M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"The next variable would be bus numbers."},{"Start":"00:03.075 ","End":"00:07.800","Text":"Bus numbers can have values such as 321,"},{"Start":"00:07.800 ","End":"00:11.820","Text":"500, 012,"},{"Start":"00:11.820 ","End":"00:14.020","Text":"and so on and so forth."},{"Start":"00:14.030 ","End":"00:16.320","Text":"The bus numbers,"},{"Start":"00:16.320 ","End":"00:20.595","Text":"they represent only the names of the routes that the bus take."},{"Start":"00:20.595 ","End":"00:24.210","Text":"So we have here a nominal variable."},{"Start":"00:24.210 ","End":"00:28.530","Text":"There\u0027s no quantitative meaning to these numbers whatsoever."},{"Start":"00:28.530 ","End":"00:31.470","Text":"There\u0027s no meaning of more or less."},{"Start":"00:31.470 ","End":"00:38.685","Text":"Bus number 500 isn\u0027t more or less than bus number 321."},{"Start":"00:38.685 ","End":"00:42.966","Text":"There is no meaning in even saying this sentence,"},{"Start":"00:42.966 ","End":"00:50.100","Text":"so here we have a nominal variable where the bus numbers are just names."},{"Start":"00:50.710 ","End":"00:54.140","Text":"The next variable is place of residence."},{"Start":"00:54.140 ","End":"00:57.005","Text":"Place of residence could be city for example."},{"Start":"00:57.005 ","End":"00:58.910","Text":"We have, let\u0027s say as an example,"},{"Start":"00:58.910 ","End":"01:01.220","Text":"Los Angeles, we could have New York,"},{"Start":"01:01.220 ","End":"01:03.550","Text":"we could have Miami,"},{"Start":"01:03.550 ","End":"01:06.155","Text":"we could have Chicago,"},{"Start":"01:06.155 ","End":"01:07.795","Text":"and so on and so forth."},{"Start":"01:07.795 ","End":"01:09.865","Text":"As we can see, these are names."},{"Start":"01:09.865 ","End":"01:13.375","Text":"There\u0027s no meaning to ordering the names."},{"Start":"01:13.375 ","End":"01:17.100","Text":"We don\u0027t have any interval characteristics."},{"Start":"01:17.100 ","End":"01:21.297","Text":"If we subtract Miami from New York there\u0027s no meaning to the difference,"},{"Start":"01:21.297 ","End":"01:23.930","Text":"there\u0027s no meaning to the operation."},{"Start":"01:23.930 ","End":"01:29.765","Text":"Obviously, we don\u0027t also have a 0 test here because there is no 0,"},{"Start":"01:29.765 ","End":"01:32.470","Text":"there is no absence of something."},{"Start":"01:32.470 ","End":"01:37.170","Text":"This then is a nominal variable."},{"Start":"01:37.170 ","End":"01:39.855","Text":"The next variable is gender."},{"Start":"01:39.855 ","End":"01:42.110","Text":"We have here male and female,"},{"Start":"01:42.110 ","End":"01:43.985","Text":"and we\u0027ve also coded them."},{"Start":"01:43.985 ","End":"01:48.130","Text":"1 is the code for male and 2 is the code for female."},{"Start":"01:48.130 ","End":"01:52.340","Text":"This is obviously a nominal variable. Why is that?"},{"Start":"01:52.340 ","End":"01:54.560","Text":"Because 1 and 2,"},{"Start":"01:54.560 ","End":"01:56.540","Text":"the codes for male and female,"},{"Start":"01:56.540 ","End":"02:03.725","Text":"don\u0027t have any meaning other than to identify 1 as being male and 2 as being female."},{"Start":"02:03.725 ","End":"02:07.050","Text":"There are no ordinal characteristics here."},{"Start":"02:07.050 ","End":"02:10.880","Text":"There\u0027s no order to male and female, to 1 and 2."},{"Start":"02:10.880 ","End":"02:13.430","Text":"There is no interval characteristics."},{"Start":"02:13.430 ","End":"02:16.010","Text":"2 minus 1 doesn\u0027t make any sense."},{"Start":"02:16.010 ","End":"02:20.875","Text":"Obviously this can\u0027t pass the 0 test because there is no 0."},{"Start":"02:20.875 ","End":"02:23.630","Text":"We don\u0027t have an absence of something."},{"Start":"02:23.630 ","End":"02:25.840","Text":"You\u0027re either male or female."},{"Start":"02:25.840 ","End":"02:29.945","Text":"Therefore, this variable is a nominal variable."},{"Start":"02:29.945 ","End":"02:32.960","Text":"The next variable is shoe size."},{"Start":"02:32.960 ","End":"02:35.420","Text":"How do we measure this size?"},{"Start":"02:35.420 ","End":"02:39.005","Text":"The shoes can have sizes in the US."},{"Start":"02:39.005 ","End":"02:42.200","Text":"The sizes could be 6, 7, 8,"},{"Start":"02:42.200 ","End":"02:43.595","Text":"and so on and so forth,"},{"Start":"02:43.595 ","End":"02:46.870","Text":"10, 12, and larger."},{"Start":"02:46.870 ","End":"02:48.455","Text":"What about in Europe?"},{"Start":"02:48.455 ","End":"02:50.870","Text":"In Europe that could be 38,"},{"Start":"02:50.870 ","End":"02:54.320","Text":"40, 42, and so on and so forth."},{"Start":"02:54.320 ","End":"02:57.125","Text":"Here, we see nominal characteristics for"},{"Start":"02:57.125 ","End":"03:00.880","Text":"each number here that represents a certain size."},{"Start":"03:00.880 ","End":"03:03.210","Text":"What about ordinal characteristics?"},{"Start":"03:03.210 ","End":"03:07.275","Text":"Yes, we know that 42 is larger than 40,"},{"Start":"03:07.275 ","End":"03:09.720","Text":"10 is larger than 8."},{"Start":"03:09.720 ","End":"03:11.835","Text":"What about interval?"},{"Start":"03:11.835 ","End":"03:17.070","Text":"Is there any meaning to 8 minus 7 or 42 minus 40?"},{"Start":"03:17.070 ","End":"03:19.335","Text":"No, and why is that?"},{"Start":"03:19.335 ","End":"03:23.390","Text":"We know that different companies create different size shoes,"},{"Start":"03:23.390 ","End":"03:29.315","Text":"but they still have the same shoe size."},{"Start":"03:29.315 ","End":"03:34.295","Text":"10 in 1 company would be a different size than 10 in another company."},{"Start":"03:34.295 ","End":"03:38.420","Text":"Therefore, the difference doesn\u0027t make any sense."},{"Start":"03:38.420 ","End":"03:40.675","Text":"What about ratio?"},{"Start":"03:40.675 ","End":"03:44.865","Text":"Shoe size can\u0027t pass the 0 test."},{"Start":"03:44.865 ","End":"03:52.120","Text":"A company can certainly have a very small shoe size that they\u0027ll call 0."},{"Start":"03:52.120 ","End":"03:55.595","Text":"This then is not the absence of shoe size,"},{"Start":"03:55.595 ","End":"03:57.770","Text":"it\u0027s not the absence of shoes,"},{"Start":"03:57.770 ","End":"04:01.610","Text":"it\u0027s just a different name for a very small shoe."},{"Start":"04:01.610 ","End":"04:06.780","Text":"So what we have here is an ordinal variable."}],"ID":13296},{"Watched":false,"Name":"Exercise 2","Duration":"3m 36s","ChapterTopicVideoID":12817,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12817.jpeg","UploadDate":"2018-07-30T17:13:18.1170000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this question, we\u0027re given the probability distribution of the number of"},{"Start":"00:03.780 ","End":"00:07.845","Text":"times that the employees of the Star Company were late getting to work,"},{"Start":"00:07.845 ","End":"00:09.615","Text":"this table right here."},{"Start":"00:09.615 ","End":"00:12.660","Text":"The company has 200 employees and we\u0027re asked,"},{"Start":"00:12.660 ","End":"00:15.240","Text":"what\u0027s the variable that\u0027s involved here?"},{"Start":"00:15.240 ","End":"00:17.340","Text":"Well, in theoretical statistics,"},{"Start":"00:17.340 ","End":"00:21.030","Text":"we\u0027re interested in analyzing groups of people."},{"Start":"00:21.030 ","End":"00:25.785","Text":"We collect data and we present the data,"},{"Start":"00:25.785 ","End":"00:27.735","Text":"and then we analyze the data."},{"Start":"00:27.735 ","End":"00:30.570","Text":"In order to collect the data,"},{"Start":"00:30.570 ","End":"00:32.910","Text":"we have to collect the data from what?"},{"Start":"00:32.910 ","End":"00:34.920","Text":"Well, from the members of the group."},{"Start":"00:34.920 ","End":"00:39.704","Text":"The members of this group are the 200 employees,"},{"Start":"00:39.704 ","End":"00:43.795","Text":"the grouping, the employees in the Star Company."},{"Start":"00:43.795 ","End":"00:47.390","Text":"What\u0027s the characteristic that we\u0027re interested in?"},{"Start":"00:47.390 ","End":"00:52.520","Text":"The characteristic is the number of times that an employee was late."},{"Start":"00:52.520 ","End":"00:56.540","Text":"This is the thing that varies from employee to employee,"},{"Start":"00:56.540 ","End":"01:02.010","Text":"so therefore this then would be the variable that\u0027s involved here."},{"Start":"01:02.010 ","End":"01:04.220","Text":"Here, the number of employees, well,"},{"Start":"01:04.220 ","End":"01:09.365","Text":"that equals to the frequency of employees having this characteristic."},{"Start":"01:09.365 ","End":"01:14.800","Text":"We can see that 17 out of 200 employees were not late at all,"},{"Start":"01:14.800 ","End":"01:18.765","Text":"23 of the 200 employees were late 1 time,"},{"Start":"01:18.765 ","End":"01:21.240","Text":"85 were late 2 times,"},{"Start":"01:21.240 ","End":"01:23.350","Text":"and so on and so forth."},{"Start":"01:23.550 ","End":"01:28.735","Text":"In this section, we\u0027re asked whether the variable is qualitative or quantitative?"},{"Start":"01:28.735 ","End":"01:30.160","Text":"If it\u0027s quantitative,"},{"Start":"01:30.160 ","End":"01:32.080","Text":"is it discrete or continuous?"},{"Start":"01:32.080 ","End":"01:33.610","Text":"Then we\u0027re asked,"},{"Start":"01:33.610 ","End":"01:35.965","Text":"what\u0027s the measuring scale of the variable?"},{"Start":"01:35.965 ","End":"01:38.500","Text":"Let\u0027s take a look at this 1 at a time."},{"Start":"01:38.500 ","End":"01:40.810","Text":"Is the variable qualitative or quantitative?"},{"Start":"01:40.810 ","End":"01:43.210","Text":"Well, it is quantitative."},{"Start":"01:43.210 ","End":"01:46.300","Text":"We are using numbers, first of all,"},{"Start":"01:46.300 ","End":"01:50.005","Text":"to represent the number of times that a person was late."},{"Start":"01:50.005 ","End":"01:53.590","Text":"There is a meaning of more or less."},{"Start":"01:53.590 ","End":"01:57.530","Text":"There is a quantitative context to these numbers."},{"Start":"01:57.530 ","End":"02:01.405","Text":"2 means that a person was late twice,"},{"Start":"02:01.405 ","End":"02:03.880","Text":"and that\u0027s more than 1,"},{"Start":"02:03.880 ","End":"02:06.940","Text":"which means that a person was late only 1 time."},{"Start":"02:06.940 ","End":"02:10.850","Text":"We have here a quantitative variable."},{"Start":"02:10.850 ","End":"02:14.690","Text":"If it\u0027s quantitative, is it discrete or continuous?"},{"Start":"02:14.690 ","End":"02:17.070","Text":"Well, obviously this is a discrete variable."},{"Start":"02:17.070 ","End":"02:23.660","Text":"Why is that? We have values from a finite set of values here."},{"Start":"02:23.660 ","End":"02:25.190","Text":"We only have 5 values,"},{"Start":"02:25.190 ","End":"02:27.605","Text":"0, 1, 2, 3, and 4."},{"Start":"02:27.605 ","End":"02:34.280","Text":"There\u0027s no infinite number of values in this scale right here."},{"Start":"02:34.280 ","End":"02:40.505","Text":"I mean, we can\u0027t have an infinite number of times that a person is late between 1 and 2."},{"Start":"02:40.505 ","End":"02:43.535","Text":"A person can\u0027t be late 1.3 times,"},{"Start":"02:43.535 ","End":"02:46.895","Text":"so the value is discrete."},{"Start":"02:46.895 ","End":"02:49.745","Text":"What\u0027s the measuring scale here?"},{"Start":"02:49.745 ","End":"02:53.120","Text":"The measuring scale is a ratio."},{"Start":"02:53.120 ","End":"02:54.890","Text":"Why is that?"},{"Start":"02:54.890 ","End":"02:58.490","Text":"Let\u0027s take a look at the 0 test."},{"Start":"02:58.490 ","End":"03:02.075","Text":"If the number of times late is 0,"},{"Start":"03:02.075 ","End":"03:09.125","Text":"that means that these people were not late at all for anytime whatsoever."},{"Start":"03:09.125 ","End":"03:12.265","Text":"There\u0027s an absence of times late."},{"Start":"03:12.265 ","End":"03:17.040","Text":"If 0 represents an absence of something,"},{"Start":"03:17.040 ","End":"03:19.515","Text":"then this then is the ratio."},{"Start":"03:19.515 ","End":"03:23.495","Text":"Also, we can see that if a person was late 4 times,"},{"Start":"03:23.495 ","End":"03:28.115","Text":"then that\u0027s twice as much as if the person was late twice."},{"Start":"03:28.115 ","End":"03:36.360","Text":"There we go. This is a quantitative discrete variable and it\u0027s a ratio variable."}],"ID":13297},{"Watched":false,"Name":"Exercise 3","Duration":"2m 22s","ChapterTopicVideoID":12818,"CourseChapterTopicPlaylistID":265039,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12818.jpeg","UploadDate":"2022-08-04T11:45:56.2430000","DurationForVideoObject":"PT2M22S","Description":null,"VideoComments":[],"Subtitles":[],"ID":13298}],"Thumbnail":null,"ID":265039},{"Name":"Summation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 49s","ChapterTopicVideoID":12820,"CourseChapterTopicPlaylistID":265040,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12820.jpeg","UploadDate":"2018-07-30T17:24:14.1530000","DurationForVideoObject":"PT3M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this chapter, I\u0027ll be talking about descriptive statistics, specifically summation."},{"Start":"00:04.410 ","End":"00:05.640","Text":"Now, in statistics,"},{"Start":"00:05.640 ","End":"00:09.090","Text":"there\u0027s an accepted way of writing the sum of observations."},{"Start":"00:09.090 ","End":"00:11.160","Text":"Let\u0027s just take a look at an example here."},{"Start":"00:11.160 ","End":"00:13.470","Text":"We have here a table with 2 columns."},{"Start":"00:13.470 ","End":"00:17.375","Text":"The first column is the column of the indices."},{"Start":"00:17.375 ","End":"00:19.725","Text":"Now, what\u0027s the index here?"},{"Start":"00:19.725 ","End":"00:21.818","Text":"Well, that\u0027s the values right here."},{"Start":"00:21.818 ","End":"00:23.685","Text":"That\u0027s 1, 2, 3, 4, 5."},{"Start":"00:23.685 ","End":"00:26.357","Text":"Now, this column is denoted by the small letter i."},{"Start":"00:26.357 ","End":"00:31.302","Text":"The second column is the column of the observations,"},{"Start":"00:31.302 ","End":"00:39.725","Text":"so x_i would be the observation where i equals a specific value."},{"Start":"00:39.725 ","End":"00:44.150","Text":"The index here is just a placeholder for the observation."},{"Start":"00:44.150 ","End":"00:48.785","Text":"If I say that I want to know what x_1 is, for example, well,"},{"Start":"00:48.785 ","End":"00:52.730","Text":"I have to look at where i equals 1 and then take a look at"},{"Start":"00:52.730 ","End":"00:58.190","Text":"the value of the observation where i equals 1, that\u0027s x_1."},{"Start":"00:58.190 ","End":"01:01.675","Text":"Now that equals to 5. Now, what about x_2?"},{"Start":"01:01.675 ","End":"01:03.080","Text":"Well, that\u0027s right here."},{"Start":"01:03.080 ","End":"01:06.650","Text":"That equals to 0. What about x_4?"},{"Start":"01:06.650 ","End":"01:08.780","Text":"Well, let\u0027s take a look at this."},{"Start":"01:08.780 ","End":"01:10.250","Text":"Well, where i equals 4,"},{"Start":"01:10.250 ","End":"01:12.070","Text":"that\u0027s this thing right here."},{"Start":"01:12.070 ","End":"01:14.745","Text":"Now, what\u0027s the observation,"},{"Start":"01:14.745 ","End":"01:17.355","Text":"x_4 where i equals 4."},{"Start":"01:17.355 ","End":"01:19.350","Text":"That equals to 3."},{"Start":"01:19.350 ","End":"01:20.940","Text":"What about x_5?"},{"Start":"01:20.940 ","End":"01:24.105","Text":"Well, again, that equals to 2."},{"Start":"01:24.105 ","End":"01:27.545","Text":"Now, having said that,"},{"Start":"01:27.545 ","End":"01:30.560","Text":"let\u0027s see how we can sum these things up."},{"Start":"01:30.560 ","End":"01:32.690","Text":"Well, how do I sum these observations?"},{"Start":"01:32.690 ","End":"01:36.470","Text":"How do I write the sum of the observations?"},{"Start":"01:36.470 ","End":"01:39.220","Text":"Well, I use this sign right here,"},{"Start":"01:39.220 ","End":"01:41.780","Text":"that\u0027s this sign right here."},{"Start":"01:41.780 ","End":"01:46.610","Text":"Now, this is the big letter Sigma in the Greek alphabet."},{"Start":"01:46.610 ","End":"01:49.158","Text":"All this is, is an operator."},{"Start":"01:49.158 ","End":"01:51.035","Text":"This is an operator that says,"},{"Start":"01:51.035 ","End":"01:53.165","Text":"\"I want to sum something.\""},{"Start":"01:53.165 ","End":"01:55.280","Text":"Now, what do I want to sum?"},{"Start":"01:55.280 ","End":"01:57.050","Text":"Well, I want to sum the observations,"},{"Start":"01:57.050 ","End":"01:58.955","Text":"so I want to sum x_i."},{"Start":"01:58.955 ","End":"02:02.570","Text":"Now, this isn\u0027t specific enough because I"},{"Start":"02:02.570 ","End":"02:06.155","Text":"want to know which of the observations need to be summed."},{"Start":"02:06.155 ","End":"02:12.365","Text":"Now, for that, I need to specify the observations using the indices right here."},{"Start":"02:12.365 ","End":"02:16.415","Text":"That means I have to say that if I want to sum up all the observations,"},{"Start":"02:16.415 ","End":"02:23.159","Text":"what that means that I need for i to start at 1 and go to 5."},{"Start":"02:23.159 ","End":"02:25.940","Text":"Where n equals 5."},{"Start":"02:25.940 ","End":"02:28.370","Text":"Now, n is the number of observations that I have,"},{"Start":"02:28.370 ","End":"02:29.900","Text":"the total number of observations,"},{"Start":"02:29.900 ","End":"02:31.310","Text":"and in our case, well,"},{"Start":"02:31.310 ","End":"02:33.620","Text":"n equals 5 because I have 5 observations."},{"Start":"02:33.620 ","End":"02:35.945","Text":"Now, how do I read this expression?"},{"Start":"02:35.945 ","End":"02:39.015","Text":"Well, that\u0027s the sum of x_i,"},{"Start":"02:39.015 ","End":"02:41.160","Text":"where i goes from 1-5."},{"Start":"02:41.160 ","End":"02:49.965","Text":"Now, that equals to x_1 plus x_2 plus x_3 plus x_4 plus x_5."},{"Start":"02:49.965 ","End":"02:52.320","Text":"Now, x_1, that equals to 5,"},{"Start":"02:52.320 ","End":"02:57.225","Text":"plus 0 plus 1 plus 3 plus 2."},{"Start":"02:57.225 ","End":"02:59.880","Text":"Now that equals to 11."},{"Start":"02:59.880 ","End":"03:05.960","Text":"Now, what happens if we want to sum a subset of the observations?"},{"Start":"03:05.960 ","End":"03:09.500","Text":"For example, these observations right here?"},{"Start":"03:09.500 ","End":"03:12.440","Text":"Well, how do I write that?"},{"Start":"03:12.440 ","End":"03:16.130","Text":"Again, I use this operator right here."},{"Start":"03:16.130 ","End":"03:17.465","Text":"That\u0027ll be the sum,"},{"Start":"03:17.465 ","End":"03:19.520","Text":"and I\u0027m summing the observations,"},{"Start":"03:19.520 ","End":"03:23.090","Text":"but now I\u0027m summing a subset of the observations,"},{"Start":"03:23.090 ","End":"03:27.685","Text":"so here i now goes from 2-4."},{"Start":"03:27.685 ","End":"03:31.723","Text":"You can read this as the sum of x_i,"},{"Start":"03:31.723 ","End":"03:33.525","Text":"where i goes from 2-4,"},{"Start":"03:33.525 ","End":"03:38.475","Text":"that equals to x_2 plus x_3 plus x_4,"},{"Start":"03:38.475 ","End":"03:42.815","Text":"and that equals to 0 plus 1 plus 3,"},{"Start":"03:42.815 ","End":"03:44.890","Text":"and that equals to 4."},{"Start":"03:44.890 ","End":"03:47.190","Text":"I think you got the idea right now,"},{"Start":"03:47.190 ","End":"03:49.870","Text":"so let\u0027s do some exercises."}],"ID":13299},{"Watched":false,"Name":"Exercise 1","Duration":"5m 22s","ChapterTopicVideoID":12821,"CourseChapterTopicPlaylistID":265040,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12821.jpeg","UploadDate":"2018-07-30T17:24:28.7600000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"In this question, we\u0027re given 5 apartments in a building."},{"Start":"00:03.225 ","End":"00:06.765","Text":"Now, the number of rooms in each apartment; that\u0027s the variable X,"},{"Start":"00:06.765 ","End":"00:09.090","Text":"and the number of people living in the apartment;"},{"Start":"00:09.090 ","End":"00:12.090","Text":"that\u0027s Y, are listed in the following table."},{"Start":"00:12.090 ","End":"00:13.485","Text":"We have X;"},{"Start":"00:13.485 ","End":"00:18.330","Text":"that\u0027s the number of rooms in the apartment,"},{"Start":"00:18.330 ","End":"00:23.550","Text":"and Y; this is the number of people living in the apartment."},{"Start":"00:23.550 ","End":"00:26.114","Text":"Now, here we have the apartment number."},{"Start":"00:26.114 ","End":"00:27.450","Text":"Now, what is that?"},{"Start":"00:27.450 ","End":"00:30.840","Text":"Well, the apartment number is actually the index,"},{"Start":"00:30.840 ","End":"00:32.685","Text":"so we have apartment number 1,"},{"Start":"00:32.685 ","End":"00:34.710","Text":"2, 3, 4, and 5."},{"Start":"00:34.710 ","End":"00:37.335","Text":"We\u0027re asked to calculate these sums."},{"Start":"00:37.335 ","End":"00:38.995","Text":"Let\u0027s get to work."},{"Start":"00:38.995 ","End":"00:42.140","Text":"The first sum is the sum of X_i."},{"Start":"00:42.140 ","End":"00:45.545","Text":"We\u0027re summing up the number of rooms,"},{"Start":"00:45.545 ","End":"00:48.245","Text":"but where i equals 1-3."},{"Start":"00:48.245 ","End":"00:52.295","Text":"That means that we have to sum up X1;"},{"Start":"00:52.295 ","End":"00:55.230","Text":"that\u0027s 2, plus X_2;"},{"Start":"00:55.230 ","End":"00:59.025","Text":"that\u0027s 3, plus X_3; that\u0027s 2."},{"Start":"00:59.025 ","End":"01:01.005","Text":"Now, that equals to 7."},{"Start":"01:01.005 ","End":"01:03.480","Text":"What about this sum right here?"},{"Start":"01:03.480 ","End":"01:05.340","Text":"Well, that\u0027s the sum of Y_i,"},{"Start":"01:05.340 ","End":"01:09.080","Text":"where i runs from 1-5."},{"Start":"01:09.080 ","End":"01:14.390","Text":"We\u0027re looking to sum up a number of people in all the apartments."},{"Start":"01:14.390 ","End":"01:18.405","Text":"Well, that equals to Y sub 1,"},{"Start":"01:18.405 ","End":"01:21.500","Text":"Y_1 that equals to 1 plus Y_2,"},{"Start":"01:21.500 ","End":"01:24.305","Text":"that\u0027s 1 plus Y_3; that\u0027s 2,"},{"Start":"01:24.305 ","End":"01:28.940","Text":"plus Y_4; that\u0027s 3 plus Y_5; that\u0027s 2."},{"Start":"01:28.940 ","End":"01:31.005","Text":"Now, that equals to 1,"},{"Start":"01:31.005 ","End":"01:34.365","Text":"2, 4, 7, 9."},{"Start":"01:34.365 ","End":"01:36.929","Text":"What about the sum?"},{"Start":"01:36.929 ","End":"01:42.195","Text":"Again, here we\u0027re summing up rooms of X_i,"},{"Start":"01:42.195 ","End":"01:44.870","Text":"where i goes from 1-4."},{"Start":"01:44.870 ","End":"01:51.050","Text":"Here we\u0027re summing up the number of rooms in the first 4 apartments."},{"Start":"01:51.050 ","End":"01:56.465","Text":"Well, that equals to 2 plus 3 plus 2 plus 4."},{"Start":"01:56.465 ","End":"01:58.265","Text":"So 2 plus 3 is 5,"},{"Start":"01:58.265 ","End":"02:02.525","Text":"plus 2 is 7, plus 4 that equals to 11."},{"Start":"02:02.525 ","End":"02:04.685","Text":"Here we have something else."},{"Start":"02:04.685 ","End":"02:07.555","Text":"We have the sum of X_i,"},{"Start":"02:07.555 ","End":"02:09.870","Text":"where i goes from 1-4,"},{"Start":"02:09.870 ","End":"02:13.260","Text":"but the sum is squared."},{"Start":"02:13.260 ","End":"02:14.610","Text":"What do we do first?"},{"Start":"02:14.610 ","End":"02:19.590","Text":"What we first do what\u0027s inside the brackets."},{"Start":"02:19.590 ","End":"02:22.605","Text":"That means that we have to sum up X_i."},{"Start":"02:22.605 ","End":"02:24.360","Text":"i goes from 1-4."},{"Start":"02:24.360 ","End":"02:25.710","Text":"Well, we did that right here,"},{"Start":"02:25.710 ","End":"02:30.559","Text":"so that\u0027ll be 2 plus 3 plus 2 plus 4."},{"Start":"02:30.559 ","End":"02:34.110","Text":"But now we have to take this sum and square it."},{"Start":"02:34.110 ","End":"02:35.950","Text":"That sum was 11,"},{"Start":"02:35.950 ","End":"02:41.385","Text":"so 11 squared that equals to 121."},{"Start":"02:41.385 ","End":"02:44.450","Text":"Let\u0027s take a look at this sum right here."},{"Start":"02:44.450 ","End":"02:46.385","Text":"Here we have the sum of X_i,"},{"Start":"02:46.385 ","End":"02:51.635","Text":"but we\u0027re not given what\u0027s the range of i."},{"Start":"02:51.635 ","End":"02:59.150","Text":"Whenever we have this form right here where we\u0027re not given what the values of i are,"},{"Start":"02:59.150 ","End":"03:05.095","Text":"then we have to assume that we\u0027re asked to sum all of the observations."},{"Start":"03:05.095 ","End":"03:08.895","Text":"That means that I have to sum all of the Xs."},{"Start":"03:08.895 ","End":"03:14.240","Text":"Well, that equals to 2 plus 3 plus 2 plus 4 plus 3."},{"Start":"03:14.240 ","End":"03:16.190","Text":"That\u0027s all 5 apartments."},{"Start":"03:16.190 ","End":"03:18.950","Text":"That\u0027s the number of rooms in all 5 apartments."},{"Start":"03:18.950 ","End":"03:21.295","Text":"That equals to 5,"},{"Start":"03:21.295 ","End":"03:25.470","Text":"7, 11, 14."},{"Start":"03:25.470 ","End":"03:28.965","Text":"That\u0027s that. Let\u0027s continue."},{"Start":"03:28.965 ","End":"03:34.065","Text":"Here we have the sum of X_i times Y_i. What comes first?"},{"Start":"03:34.065 ","End":"03:40.760","Text":"Well, obviously, the multiplication within the sum that\u0027s what comes first."},{"Start":"03:40.760 ","End":"03:48.150","Text":"Here we\u0027re looking at X_1 times Y_1 plus so on and so forth."},{"Start":"03:48.150 ","End":"03:50.820","Text":"But since i isn\u0027t specified,"},{"Start":"03:50.820 ","End":"03:53.780","Text":"then we have to do this for all the apartments."},{"Start":"03:53.780 ","End":"03:55.610","Text":"That\u0027ll be X_1;"},{"Start":"03:55.610 ","End":"03:57.410","Text":"that\u0027s 2 times Y_1,"},{"Start":"03:57.410 ","End":"04:03.150","Text":"that\u0027s 1, plus 3 times 1;"},{"Start":"04:03.150 ","End":"04:05.700","Text":"that\u0027s X_2 and Y_2,"},{"Start":"04:05.700 ","End":"04:07.874","Text":"plus X_3 and Y_3; well,"},{"Start":"04:07.874 ","End":"04:12.810","Text":"that\u0027s 2 times 2 plus X_4 and Y_4,"},{"Start":"04:12.810 ","End":"04:14.730","Text":"that will be 4 times 3,"},{"Start":"04:14.730 ","End":"04:19.005","Text":"plus X_5 and Y_5;"},{"Start":"04:19.005 ","End":"04:21.750","Text":"that\u0027ll be 3 times 2."},{"Start":"04:21.750 ","End":"04:25.935","Text":"Now, that equals to 27."},{"Start":"04:25.935 ","End":"04:28.890","Text":"What happens here? Here,"},{"Start":"04:28.890 ","End":"04:30.440","Text":"this is something, again,"},{"Start":"04:30.440 ","End":"04:31.700","Text":"totally different than here."},{"Start":"04:31.700 ","End":"04:38.870","Text":"Here we have the sum of the multiplication and here we have a multiplication of the sum."},{"Start":"04:38.870 ","End":"04:42.965","Text":"Here what we need to do is we need to sum up all the Xs,"},{"Start":"04:42.965 ","End":"04:45.390","Text":"we need to sum up all the Ys,"},{"Start":"04:45.390 ","End":"04:47.960","Text":"and then multiply the results together."},{"Start":"04:47.960 ","End":"04:52.135","Text":"That will be the sum of all the Xs,"},{"Start":"04:52.135 ","End":"05:00.270","Text":"that\u0027s 2 plus 3 plus 2 plus 4 plus 3 times,"},{"Start":"05:00.270 ","End":"05:03.090","Text":"now, the summing up all the Ys."},{"Start":"05:03.090 ","End":"05:09.630","Text":"That\u0027ll be 1 plus 1 plus 2 plus 3 plus 2."},{"Start":"05:09.630 ","End":"05:14.130","Text":"Now, that comes out to 126."},{"Start":"05:14.130 ","End":"05:22.140","Text":"This then is a small exercise in how we sum the observations."}],"ID":13300},{"Watched":false,"Name":"Exercise 2","Duration":"5m 7s","ChapterTopicVideoID":12822,"CourseChapterTopicPlaylistID":265040,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12822.jpeg","UploadDate":"2018-07-30T17:24:45.6470000","DurationForVideoObject":"PT5M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this question, we\u0027re given a table that shows"},{"Start":"00:02.250 ","End":"00:05.055","Text":"the value for the variables x_i and y_i,"},{"Start":"00:05.055 ","End":"00:07.230","Text":"where i goes from 1-6."},{"Start":"00:07.230 ","End":"00:12.950","Text":"Now, assume also that we have constants that a equals to 2 and b equals to 5,"},{"Start":"00:12.950 ","End":"00:17.365","Text":"and we\u0027re asked to calculate all of the following formulas."},{"Start":"00:17.365 ","End":"00:23.300","Text":"Here we have the table and we have our variables x_i and y_i,"},{"Start":"00:23.300 ","End":"00:25.025","Text":"and here we have our index."},{"Start":"00:25.025 ","End":"00:26.780","Text":"Our index is 1, 2, 3,"},{"Start":"00:26.780 ","End":"00:29.010","Text":"4, 5, and 6, and as we said,"},{"Start":"00:29.010 ","End":"00:32.700","Text":"the index is just a place holder for the observations,"},{"Start":"00:32.700 ","End":"00:38.580","Text":"and here we have our observations for x and our observations for y."},{"Start":"00:38.580 ","End":"00:41.060","Text":"We can see, for example,"},{"Start":"00:41.060 ","End":"00:46.906","Text":"that x_3 for example,"},{"Start":"00:46.906 ","End":"00:51.170","Text":"we go to the third index right here where i equals 3 and we"},{"Start":"00:51.170 ","End":"00:56.300","Text":"go to the observation for x and that equals to 4."},{"Start":"00:56.300 ","End":"01:00.300","Text":"What about y_5?"},{"Start":"01:00.300 ","End":"01:03.750","Text":"Well, we go to the fifth index i equals 5 and go"},{"Start":"01:03.750 ","End":"01:09.585","Text":"the observation for y that equals to minus 5."},{"Start":"01:09.585 ","End":"01:14.570","Text":"We also have constants a equals 2 and b equals 5."},{"Start":"01:14.570 ","End":"01:16.460","Text":"Now let\u0027s see what we can do here."},{"Start":"01:16.460 ","End":"01:19.025","Text":"Let\u0027s calculate the following formulas."},{"Start":"01:19.025 ","End":"01:27.435","Text":"Well, we have a, that\u0027s the sum where i goes from 1-4 of y_i."},{"Start":"01:27.435 ","End":"01:33.905","Text":"First of all, we\u0027re looking at only 4 indices right here where i equals 1,"},{"Start":"01:33.905 ","End":"01:35.520","Text":"2, 3, and 4,"},{"Start":"01:35.520 ","End":"01:39.125","Text":"and we\u0027re summing up the y values, the y observations."},{"Start":"01:39.125 ","End":"01:43.490","Text":"That would be equal to y_1 that\u0027s 2,"},{"Start":"01:43.490 ","End":"01:47.960","Text":"plus Y_2 that\u0027s 0, plus Y_3,"},{"Start":"01:47.960 ","End":"01:51.945","Text":"that\u0027s 0 and Y_4 that equals to 1,"},{"Start":"01:51.945 ","End":"01:54.900","Text":"so that equals to 3."},{"Start":"01:54.900 ","End":"01:57.890","Text":"What about this thing right here?"},{"Start":"01:57.890 ","End":"02:03.035","Text":"That\u0027s the sum where i goes from 1 to 6 of a constant a."},{"Start":"02:03.035 ","End":"02:04.995","Text":"What do we have to do?"},{"Start":"02:04.995 ","End":"02:08.480","Text":"We have to take the value a, that\u0027s 2."},{"Start":"02:08.480 ","End":"02:11.315","Text":"Now, we have to sum that up how many times?"},{"Start":"02:11.315 ","End":"02:15.605","Text":"6 times so that\u0027s 2 plus 2 plus 2,"},{"Start":"02:15.605 ","End":"02:19.230","Text":"plus 2 plus 2 plus 2. Why is that?"},{"Start":"02:19.230 ","End":"02:23.130","Text":"Because i runs from 1 to 6,"},{"Start":"02:23.130 ","End":"02:27.570","Text":"so we have 6 values of a right here,"},{"Start":"02:27.570 ","End":"02:32.240","Text":"that equals to 6 times 2 and that equals to 12."},{"Start":"02:32.240 ","End":"02:39.240","Text":"Now, what about the sum where i goes from 1 to 6 of x_i times y_i."},{"Start":"02:39.240 ","End":"02:40.530","Text":"The first thing we need to do,"},{"Start":"02:40.530 ","End":"02:44.565","Text":"is we need to multiply x_1 by y_1."},{"Start":"02:44.565 ","End":"02:49.490","Text":"Then we have to add to that X_2 times Y_2 and so on and so forth."},{"Start":"02:49.490 ","End":"02:51.080","Text":"Let\u0027s do that."},{"Start":"02:51.080 ","End":"02:58.020","Text":"X_1 is 3 times Y_1 that equals to 2 plus,"},{"Start":"02:58.020 ","End":"03:01.215","Text":"now X_2 is 2,"},{"Start":"03:01.215 ","End":"03:06.900","Text":"times Y_2, that\u0027s 0 plus 4 times 0,"},{"Start":"03:06.900 ","End":"03:11.205","Text":"plus minus 2 times 1,"},{"Start":"03:11.205 ","End":"03:15.510","Text":"plus 1 times minus 5,"},{"Start":"03:15.510 ","End":"03:18.765","Text":"plus 4 times 2."},{"Start":"03:18.765 ","End":"03:22.910","Text":"That equals to 7."},{"Start":"03:22.910 ","End":"03:26.005","Text":"Now, what about this thing right here?"},{"Start":"03:26.005 ","End":"03:32.345","Text":"This is the sum from where i goes from 1 to 6 of x_i plus y_i."},{"Start":"03:32.345 ","End":"03:34.035","Text":"Now, how does that look?"},{"Start":"03:34.035 ","End":"03:38.050","Text":"We take x_1 plus y_1 and we add to"},{"Start":"03:38.050 ","End":"03:43.030","Text":"that X_2 plus Y_2 and so on and so forth, so let\u0027s do that."},{"Start":"03:43.030 ","End":"03:45.820","Text":"That\u0027ll be X_1 is 3,"},{"Start":"03:45.820 ","End":"03:48.160","Text":"plus Y_1 is 2,"},{"Start":"03:48.160 ","End":"03:53.970","Text":"plus 2 plus 0,"},{"Start":"03:53.970 ","End":"03:57.160","Text":"plus 4 plus 0,"},{"Start":"03:57.160 ","End":"04:05.490","Text":"plus minus 2, plus 1,"},{"Start":"04:05.490 ","End":"04:09.555","Text":"plus 1 plus minus 5,"},{"Start":"04:09.555 ","End":"04:13.005","Text":"plus 4 plus 2,"},{"Start":"04:13.005 ","End":"04:18.300","Text":"and that equals 12 right here."},{"Start":"04:18.300 ","End":"04:21.550","Text":"Now what about this expression right here?"},{"Start":"04:21.550 ","End":"04:23.230","Text":"Well, that\u0027s the sum of x_i,"},{"Start":"04:23.230 ","End":"04:26.230","Text":"where i goes from 1-6 plus a."},{"Start":"04:26.230 ","End":"04:28.960","Text":"Now here we see that a isn\u0027t part of"},{"Start":"04:28.960 ","End":"04:33.835","Text":"the sum operator right here so that would be equal to the following."},{"Start":"04:33.835 ","End":"04:37.330","Text":"We sum up all the x_is,"},{"Start":"04:37.330 ","End":"04:38.800","Text":"where i goes from 1-6,"},{"Start":"04:38.800 ","End":"04:42.410","Text":"so that\u0027ll be 3 plus, 2 plus,"},{"Start":"04:42.410 ","End":"04:44.360","Text":"4 plus, minus 2,"},{"Start":"04:44.360 ","End":"04:47.150","Text":"plus 1, plus 4."},{"Start":"04:47.150 ","End":"04:49.550","Text":"That\u0027s the sum of x_i,"},{"Start":"04:49.550 ","End":"04:56.170","Text":"where i goes from 1-6 plus a now a is 2,"},{"Start":"04:56.170 ","End":"05:03.690","Text":"so all that equals to 14."},{"Start":"05:03.690 ","End":"05:07.740","Text":"These are the answers for these formulas."}],"ID":13301},{"Watched":false,"Name":"Exercise 3","Duration":"3m 32s","ChapterTopicVideoID":12823,"CourseChapterTopicPlaylistID":265040,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12823.jpeg","UploadDate":"2018-07-30T17:24:56.6770000","DurationForVideoObject":"PT3M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this question we\u0027re asked to state whether the following equalities are true or false."},{"Start":"00:05.145 ","End":"00:07.290","Text":"Let\u0027s take a look at the first one,"},{"Start":"00:07.290 ","End":"00:10.755","Text":"which is the sum of b times x_i,"},{"Start":"00:10.755 ","End":"00:17.040","Text":"where i goes from 1 to n. Does that equal to b times the sum of x_i?"},{"Start":"00:17.040 ","End":"00:20.490","Text":"Let\u0027s start with this side of the equation right here."},{"Start":"00:20.490 ","End":"00:25.095","Text":"That\u0027s the sum of b times x_i,"},{"Start":"00:25.095 ","End":"00:27.090","Text":"i goes from 1 to n. Now,"},{"Start":"00:27.090 ","End":"00:28.785","Text":"b is a constant."},{"Start":"00:28.785 ","End":"00:31.040","Text":"What does that equal to?"},{"Start":"00:31.040 ","End":"00:37.250","Text":"This whole sum equals to b times x_1 plus"},{"Start":"00:37.250 ","End":"00:43.945","Text":"b times x_2 plus and so on and so forth plus b times x_n."},{"Start":"00:43.945 ","End":"00:49.575","Text":"Now let\u0027s take b out of the sum as a common factor,"},{"Start":"00:49.575 ","End":"00:58.485","Text":"so that would be b times x_1 plus x_2 and so on and so forth plus x_n."},{"Start":"00:58.485 ","End":"01:02.970","Text":"Now we can see here that x_1 plus x_2 plus and so on and so forth"},{"Start":"01:02.970 ","End":"01:07.320","Text":"till x_n that equals to the sum of x_i,"},{"Start":"01:07.320 ","End":"01:09.616","Text":"so this expression,"},{"Start":"01:09.616 ","End":"01:14.390","Text":"that equals to b times the sum of x_i,"},{"Start":"01:14.390 ","End":"01:21.120","Text":"where i goes from 1 to n. Therefore this equality is true."},{"Start":"01:21.120 ","End":"01:23.280","Text":"What about the next one,"},{"Start":"01:23.280 ","End":"01:27.105","Text":"the sum of a where i goes from 1 to n,"},{"Start":"01:27.105 ","End":"01:29.640","Text":"does that equal to a times n?"},{"Start":"01:29.640 ","End":"01:31.770","Text":"Well a is a constant."},{"Start":"01:31.770 ","End":"01:34.050","Text":"Again let\u0027s starting in this side,"},{"Start":"01:34.050 ","End":"01:41.120","Text":"the sum of a where i goes from 1 to n. That means that we have to take a"},{"Start":"01:41.120 ","End":"01:44.690","Text":"constant a and we have to sum that and"},{"Start":"01:44.690 ","End":"01:49.310","Text":"times plus a plus a and so on and so forth plus a,"},{"Start":"01:49.310 ","End":"01:54.040","Text":"and we have to do that n times."},{"Start":"01:54.260 ","End":"02:01.675","Text":"That equals to a times n. This is exactly what we have to show."},{"Start":"02:01.675 ","End":"02:05.385","Text":"Therefore this equality is true."},{"Start":"02:05.385 ","End":"02:08.650","Text":"What about these expressions right here?"},{"Start":"02:08.650 ","End":"02:10.833","Text":"Well, they\u0027re not true."},{"Start":"02:10.833 ","End":"02:13.870","Text":"Why is that? Let\u0027s take a look at an example here."},{"Start":"02:13.870 ","End":"02:16.060","Text":"Assume that x_1,"},{"Start":"02:16.060 ","End":"02:18.970","Text":"that will be equal to 3 and x_2,"},{"Start":"02:18.970 ","End":"02:20.935","Text":"that\u0027ll be equal to 4."},{"Start":"02:20.935 ","End":"02:23.530","Text":"Let\u0027s calculate this side right here."},{"Start":"02:23.530 ","End":"02:29.160","Text":"That will be the sum where i goes from 1 to 2 of x_i,"},{"Start":"02:29.160 ","End":"02:32.550","Text":"and then we have to take this sum and square it."},{"Start":"02:32.550 ","End":"02:36.045","Text":"That means that we have to take 3 plus 4,"},{"Start":"02:36.045 ","End":"02:41.985","Text":"x_1 equals 3 and x_2 equals 4 then we have to square the sum."},{"Start":"02:41.985 ","End":"02:46.185","Text":"That equals to 7 squared which equals to 49."},{"Start":"02:46.185 ","End":"02:48.740","Text":"Let\u0027s take a look at this side right here."},{"Start":"02:48.740 ","End":"02:50.095","Text":"That\u0027s the sum,"},{"Start":"02:50.095 ","End":"02:55.560","Text":"where i goes from 1 to 2 of x_i squared."},{"Start":"02:55.560 ","End":"02:58.425","Text":"That equals to 3 squared,"},{"Start":"02:58.425 ","End":"03:03.500","Text":"x_1 is 3 squared plus 4 squared."},{"Start":"03:03.500 ","End":"03:08.570","Text":"Now that equals to 9 plus 16 and that equals to 25."},{"Start":"03:08.570 ","End":"03:13.220","Text":"We see here that we do not have an equality."},{"Start":"03:13.220 ","End":"03:18.320","Text":"This expression right here means what we call that,"},{"Start":"03:18.320 ","End":"03:20.630","Text":"the square of the sums."},{"Start":"03:20.630 ","End":"03:25.010","Text":"In here, we call this expression the sum of the squares."},{"Start":"03:25.010 ","End":"03:32.130","Text":"We can see that the square of the sums does not equal to the sum of the squares."}],"ID":13302},{"Watched":false,"Name":"Exercise 4","Duration":"2m 53s","ChapterTopicVideoID":12824,"CourseChapterTopicPlaylistID":265040,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12824.jpeg","UploadDate":"2018-07-30T17:25:08.7830000","DurationForVideoObject":"PT2M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this question, we\u0027re given that the sum of x_i,"},{"Start":"00:02.670 ","End":"00:04.260","Text":"where i goes from 1-10,"},{"Start":"00:04.260 ","End":"00:05.655","Text":"well, that equals to 80."},{"Start":"00:05.655 ","End":"00:08.010","Text":"The sum of x_i squared,"},{"Start":"00:08.010 ","End":"00:09.450","Text":"where i goes from 1-10,"},{"Start":"00:09.450 ","End":"00:12.000","Text":"that equals to 1640."},{"Start":"00:12.000 ","End":"00:17.835","Text":"We\u0027re asked to calculate the sum of x_i minus 4 of that squared,"},{"Start":"00:17.835 ","End":"00:20.220","Text":"where i goes from 1-10."},{"Start":"00:20.220 ","End":"00:24.105","Text":"Now, let\u0027s just recall from basic algebra,"},{"Start":"00:24.105 ","End":"00:28.740","Text":"that we have a minus b squared, well,"},{"Start":"00:28.740 ","End":"00:34.035","Text":"that equals to a squared minus 2 a_b, plus b squared."},{"Start":"00:34.035 ","End":"00:37.995","Text":"Why do we need that? We need that to expand this expression."},{"Start":"00:37.995 ","End":"00:41.905","Text":"Let\u0027s get to work. Let\u0027s calculate this expression."},{"Start":"00:41.905 ","End":"00:44.490","Text":"That will be equal to the sum,"},{"Start":"00:44.490 ","End":"00:46.935","Text":"i goes from 1-10,"},{"Start":"00:46.935 ","End":"00:50.855","Text":"as we said, x_i minus 4 squared."},{"Start":"00:50.855 ","End":"00:52.810","Text":"Now let\u0027s expand that."},{"Start":"00:52.810 ","End":"00:58.170","Text":"i goes from 1-10 and that becomes x_i"},{"Start":"00:58.170 ","End":"01:04.960","Text":"squared minus 8 x_i plus 16."},{"Start":"01:05.090 ","End":"01:08.655","Text":"This thing now becomes the following,"},{"Start":"01:08.655 ","End":"01:16.740","Text":"that\u0027s the sum of x_i squared plus the sum of minus,"},{"Start":"01:16.740 ","End":"01:26.745","Text":"here we go, that\u0027ll be minus 8 times x_i plus the sum of 16 for all the sums,"},{"Start":"01:26.745 ","End":"01:28.485","Text":"i goes from 1-10,"},{"Start":"01:28.485 ","End":"01:30.915","Text":"i goes from 1-10,"},{"Start":"01:30.915 ","End":"01:34.840","Text":"and i goes from 1-10."},{"Start":"01:35.810 ","End":"01:37.950","Text":"What is that equal to?"},{"Start":"01:37.950 ","End":"01:41.180","Text":"Let\u0027s leave this expression here alone."},{"Start":"01:41.180 ","End":"01:45.945","Text":"That will be the sum of i equals 1-10 of x_i squared."},{"Start":"01:45.945 ","End":"01:47.465","Text":"This is a constant,"},{"Start":"01:47.465 ","End":"01:50.030","Text":"the sum of a constant times x_i."},{"Start":"01:50.030 ","End":"01:53.585","Text":"We saw before that we can take the constant out of the sum."},{"Start":"01:53.585 ","End":"01:57.900","Text":"That be minus 8 times the sum of x_i,"},{"Start":"01:57.900 ","End":"02:01.624","Text":"i goes from 1-10 plus,"},{"Start":"02:01.624 ","End":"02:04.530","Text":"here we are summing a constant,"},{"Start":"02:04.530 ","End":"02:11.470","Text":"and n here with that equals to 10 times 16."},{"Start":"02:12.140 ","End":"02:18.530","Text":"We can go ahead and plug in these values into this expression."},{"Start":"02:18.530 ","End":"02:23.130","Text":"That equals to the following that is the sum of x_i squared."},{"Start":"02:23.130 ","End":"02:29.010","Text":"That equals to 1640 minus 8 times,"},{"Start":"02:29.010 ","End":"02:30.525","Text":"well, what\u0027s the sum of x_i,"},{"Start":"02:30.525 ","End":"02:32.400","Text":"that equals to 80."},{"Start":"02:32.400 ","End":"02:35.220","Text":"Plus 10 times 16,"},{"Start":"02:35.220 ","End":"02:37.185","Text":"well, that is 160,"},{"Start":"02:37.185 ","End":"02:42.225","Text":"and all this comes out to 1160."},{"Start":"02:42.225 ","End":"02:45.390","Text":"The sum of x_i minus 4,"},{"Start":"02:45.390 ","End":"02:46.845","Text":"all of that squared,"},{"Start":"02:46.845 ","End":"02:49.095","Text":"where I goes from 1-10, well,"},{"Start":"02:49.095 ","End":"02:54.220","Text":"that equals to 1160."}],"ID":13303}],"Thumbnail":null,"ID":265040},{"Name":"Measure of Central Tendency","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"58s","ChapterTopicVideoID":12825,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12825.jpeg","UploadDate":"2018-07-30T17:26:37.3030000","DurationForVideoObject":"PT58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.029","Text":"In this chapter, I\u0027ll be talking about Descriptive Statistics,"},{"Start":"00:03.029 ","End":"00:06.075","Text":"specifically measures of central tendency."},{"Start":"00:06.075 ","End":"00:07.875","Text":"What\u0027s our goal here?"},{"Start":"00:07.875 ","End":"00:09.830","Text":"Let\u0027s think of it this way,"},{"Start":"00:09.830 ","End":"00:12.300","Text":"after collecting and presenting the data,"},{"Start":"00:12.300 ","End":"00:14.390","Text":"we need to analyze it."},{"Start":"00:14.390 ","End":"00:19.965","Text":"In this chapter, we\u0027ll deal with what\u0027s called Measures of Centrality."},{"Start":"00:19.965 ","End":"00:23.790","Text":"The purpose of these measures is to measure"},{"Start":"00:23.790 ","End":"00:28.095","Text":"the center of the probability distribution of the observations."},{"Start":"00:28.095 ","End":"00:31.290","Text":"For example, for dealing with employee salaries,"},{"Start":"00:31.290 ","End":"00:34.365","Text":"then the question that we\u0027ll ask in this chapter is,"},{"Start":"00:34.365 ","End":"00:38.000","Text":"what\u0027s the central value of the employee salaries?"},{"Start":"00:38.000 ","End":"00:41.185","Text":"We automatically think of averages,"},{"Start":"00:41.185 ","End":"00:44.015","Text":"but there are other metrics that we should consider."},{"Start":"00:44.015 ","End":"00:46.580","Text":"We\u0027ll talk about these other metrics,"},{"Start":"00:46.580 ","End":"00:48.005","Text":"the logic behind them,"},{"Start":"00:48.005 ","End":"00:52.250","Text":"and how to calculate them for each 1 of the presentation of"},{"Start":"00:52.250 ","End":"00:58.350","Text":"visualization techniques that we learned in the past chapter."}],"ID":13304},{"Watched":false,"Name":"Tutorial 2","Duration":"7m 45s","ChapterTopicVideoID":12835,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12835.jpeg","UploadDate":"2018-07-30T17:30:17.2400000","DurationForVideoObject":"PT7M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"The first technique that we\u0027ll talk about is called the mode,"},{"Start":"00:03.285 ","End":"00:05.925","Text":"and it\u0027s denoted by Mo."},{"Start":"00:05.925 ","End":"00:09.495","Text":"Now, the definition of the mode is that"},{"Start":"00:09.495 ","End":"00:13.340","Text":"it\u0027s the most common value in the probability distribution."},{"Start":"00:13.340 ","End":"00:16.170","Text":"If we\u0027re given a list of numbers,"},{"Start":"00:16.170 ","End":"00:20.655","Text":"then the mode would be the value that repeats itself the most times."},{"Start":"00:20.655 ","End":"00:25.110","Text":"Now, for example, let\u0027s say we have this list right here."},{"Start":"00:25.110 ","End":"00:27.840","Text":"Now, what\u0027s the most common value?"},{"Start":"00:27.840 ","End":"00:32.145","Text":"Well, that\u0027s 4, that\u0027s right here and that\u0027s right here,"},{"Start":"00:32.145 ","End":"00:35.970","Text":"that\u0027s the value that repeats itself the most times,"},{"Start":"00:35.970 ","End":"00:41.310","Text":"so that means that if we have this list of values,"},{"Start":"00:41.310 ","End":"00:45.730","Text":"then the mode will be equal to 4."},{"Start":"00:45.730 ","End":"00:48.290","Text":"Now, assume that we have data in"},{"Start":"00:48.290 ","End":"00:51.695","Text":"a discrete frequency tables just like this one right here."},{"Start":"00:51.695 ","End":"00:53.720","Text":"How do I identify the mode here?"},{"Start":"00:53.720 ","End":"00:57.695","Text":"Well, that\u0027s the value whose frequency is the highest."},{"Start":"00:57.695 ","End":"00:59.180","Text":"Again, in this table,"},{"Start":"00:59.180 ","End":"01:01.190","Text":"where\u0027s the highest frequency?"},{"Start":"01:01.190 ","End":"01:04.140","Text":"Well, the highest frequency is this guy right here."},{"Start":"01:04.140 ","End":"01:09.170","Text":"What\u0027s the x value that receives the highest frequency?"},{"Start":"01:09.170 ","End":"01:11.450","Text":"Well, that\u0027s 0 right here."},{"Start":"01:11.450 ","End":"01:15.545","Text":"The mode here, that\u0027ll be equal to 0."},{"Start":"01:15.545 ","End":"01:17.150","Text":"Now, we have to pay attention,"},{"Start":"01:17.150 ","End":"01:19.235","Text":"it\u0027s not 100, again,"},{"Start":"01:19.235 ","End":"01:26.000","Text":"the value 100 is the frequency of the mode,"},{"Start":"01:26.000 ","End":"01:30.008","Text":"that\u0027s the highest frequency in all of the distribution,"},{"Start":"01:30.008 ","End":"01:35.850","Text":"so the mode would be the value receiving the highest frequency."},{"Start":"01:35.980 ","End":"01:39.410","Text":"When data is presented to us in a bar chart, well,"},{"Start":"01:39.410 ","End":"01:41.780","Text":"that\u0027s very easy to identify the mode;"},{"Start":"01:41.780 ","End":"01:46.195","Text":"that\u0027s the value of the x for the highest bar."},{"Start":"01:46.195 ","End":"01:50.810","Text":"Let\u0027s say we have here our bar chart on the x-axis,"},{"Start":"01:50.810 ","End":"01:53.615","Text":"we have here marks of students,"},{"Start":"01:53.615 ","End":"01:55.310","Text":"and on the y-axis,"},{"Start":"01:55.310 ","End":"01:56.975","Text":"we have the frequency,"},{"Start":"01:56.975 ","End":"01:59.705","Text":"the number of students receiving these marks."},{"Start":"01:59.705 ","End":"02:04.980","Text":"For example, we have 2 students receiving a mark of 5,"},{"Start":"02:04.980 ","End":"02:07.800","Text":"4 students receiving a mark of 6,"},{"Start":"02:07.800 ","End":"02:10.100","Text":"8 students receiving a mark of 7,"},{"Start":"02:10.100 ","End":"02:11.330","Text":"and so on and so forth."},{"Start":"02:11.330 ","End":"02:13.310","Text":"Now, where\u0027s the highest bar?"},{"Start":"02:13.310 ","End":"02:15.230","Text":"Well, that\u0027s right here."},{"Start":"02:15.230 ","End":"02:18.245","Text":"That means our mode,"},{"Start":"02:18.245 ","End":"02:20.645","Text":"well, that has to equal to 7."},{"Start":"02:20.645 ","End":"02:27.550","Text":"Remember, again, the mode is the x value that has the highest bar,"},{"Start":"02:27.550 ","End":"02:31.460","Text":"it\u0027s not the frequency of x,"},{"Start":"02:31.460 ","End":"02:34.715","Text":"it\u0027ll be the x value itself."},{"Start":"02:34.715 ","End":"02:38.195","Text":"When the data is presented to us in a pie chart,"},{"Start":"02:38.195 ","End":"02:41.405","Text":"then the mode would be the largest slice of pie."},{"Start":"02:41.405 ","End":"02:44.930","Text":"For example, here we have a probability distribution of"},{"Start":"02:44.930 ","End":"02:49.070","Text":"marital status and the values of x would be married,"},{"Start":"02:49.070 ","End":"02:51.169","Text":"widowed, single, and divorced."},{"Start":"02:51.169 ","End":"02:52.925","Text":"Now, what\u0027s the largest slice?"},{"Start":"02:52.925 ","End":"02:55.700","Text":"Well, that\u0027s the blue slice right here, that\u0027s married,"},{"Start":"02:55.700 ","End":"02:58.760","Text":"that\u0027s the value of x that has the largest slice,"},{"Start":"02:58.760 ","End":"03:04.050","Text":"so the mode here will be married."},{"Start":"03:05.550 ","End":"03:10.555","Text":"Here, the data is presented in a frequency table by categories."},{"Start":"03:10.555 ","End":"03:13.435","Text":"Now, how do we identified the mode here?"},{"Start":"03:13.435 ","End":"03:17.845","Text":"Well, that\u0027s defined as the middle of the category that has the highest density."},{"Start":"03:17.845 ","End":"03:19.120","Text":"Now, we have to pay attention,"},{"Start":"03:19.120 ","End":"03:20.710","Text":"it\u0027s not the highest frequency,"},{"Start":"03:20.710 ","End":"03:22.630","Text":"it\u0027s the highest density and why is that?"},{"Start":"03:22.630 ","End":"03:25.460","Text":"Well, first of all, let\u0027s take a look at our example here."},{"Start":"03:25.460 ","End":"03:29.350","Text":"We have here our x values and these are categories,"},{"Start":"03:29.350 ","End":"03:34.240","Text":"and 0-60 is the boundaries of the first category,"},{"Start":"03:34.240 ","End":"03:37.000","Text":"60-70, the boundary of the second category,"},{"Start":"03:37.000 ","End":"03:38.470","Text":"and so on and so forth."},{"Start":"03:38.470 ","End":"03:41.155","Text":"Now, the highest frequency,"},{"Start":"03:41.155 ","End":"03:42.505","Text":"obviously, is 20,"},{"Start":"03:42.505 ","End":"03:45.850","Text":"right here, but is that the mode right here,"},{"Start":"03:45.850 ","End":"03:47.290","Text":"the middle of this category?"},{"Start":"03:47.290 ","End":"03:48.783","Text":"Well, obviously not."},{"Start":"03:48.783 ","End":"03:49.955","Text":"Why is that?"},{"Start":"03:49.955 ","End":"03:54.560","Text":"Well, if we take a look at the width of this category,"},{"Start":"03:54.560 ","End":"03:56.990","Text":"we\u0027ll see that the width here is 60,"},{"Start":"03:56.990 ","End":"03:59.255","Text":"60 minus 0, that\u0027s 60."},{"Start":"03:59.255 ","End":"04:01.400","Text":"But here it\u0027s 70 minus 60,"},{"Start":"04:01.400 ","End":"04:04.160","Text":"that\u0027s 10, and here it\u0027s also 10,"},{"Start":"04:04.160 ","End":"04:08.605","Text":"so we don\u0027t have equal width for all the categories,"},{"Start":"04:08.605 ","End":"04:11.389","Text":"so we can\u0027t use the frequency,"},{"Start":"04:11.389 ","End":"04:13.775","Text":"we have to use the density."},{"Start":"04:13.775 ","End":"04:19.220","Text":"If that\u0027s the case, let\u0027s just define our density for each category."},{"Start":"04:19.220 ","End":"04:21.455","Text":"Well, what\u0027s the definition of a density?"},{"Start":"04:21.455 ","End":"04:28.115","Text":"Well, the density is defined as the frequency divided by the width of the category,"},{"Start":"04:28.115 ","End":"04:31.636","Text":"that\u0027s L_H minus L_L,"},{"Start":"04:31.636 ","End":"04:38.480","Text":"the upper boundary, the higher bound minus the lower bound right here, so in our case,"},{"Start":"04:38.480 ","End":"04:40.670","Text":"that will be the density,"},{"Start":"04:40.670 ","End":"04:45.440","Text":"that\u0027ll be 20 divided by 60 minus 0,"},{"Start":"04:45.440 ","End":"04:47.750","Text":"and that equals to 1/3."},{"Start":"04:47.750 ","End":"04:52.250","Text":"Here, that\u0027ll be 10 divided by 70 minus 60,"},{"Start":"04:52.250 ","End":"04:54.635","Text":"that\u0027s 10 that equals to 1."},{"Start":"04:54.635 ","End":"05:01.145","Text":"Here, that\u0027ll be 18 divided by 10, that equals to 1.8."},{"Start":"05:01.145 ","End":"05:04.811","Text":"Here, that\u0027ll be 15 divided by 10,"},{"Start":"05:04.811 ","End":"05:07.220","Text":"that equals to 1.5."},{"Start":"05:07.220 ","End":"05:08.690","Text":"Here, that\u0027ll be the same,"},{"Start":"05:08.690 ","End":"05:10.460","Text":"that\u0027ll be 15 divided by 10,"},{"Start":"05:10.460 ","End":"05:12.565","Text":"that equals to 1.5."},{"Start":"05:12.565 ","End":"05:20.330","Text":"We can see that the highest density is right here in this category right here,"},{"Start":"05:20.330 ","End":"05:22.460","Text":"in the third category."},{"Start":"05:22.460 ","End":"05:25.205","Text":"How do we define our mode?"},{"Start":"05:25.205 ","End":"05:29.480","Text":"Well, that\u0027s the middle of the category that has the highest density."},{"Start":"05:29.480 ","End":"05:32.209","Text":"Well, this is the category with the highest density,"},{"Start":"05:32.209 ","End":"05:36.050","Text":"so we need to calculate the middle of this category."},{"Start":"05:36.050 ","End":"05:40.340","Text":"That\u0027ll be 70 plus 80 divided by 2,"},{"Start":"05:40.340 ","End":"05:43.220","Text":"and that equals to 75,"},{"Start":"05:43.220 ","End":"05:46.655","Text":"so 75 is our mode."},{"Start":"05:46.655 ","End":"05:50.180","Text":"Here, we have a histogram,"},{"Start":"05:50.180 ","End":"05:53.130","Text":"the way the data is presented to us is a histogram."},{"Start":"05:53.130 ","End":"05:54.400","Text":"Now, what\u0027s a histogram?"},{"Start":"05:54.400 ","End":"05:59.270","Text":"Well, that\u0027s just a graphical representation of a frequency table with categories."},{"Start":"05:59.270 ","End":"06:01.325","Text":"How do we identify the mode here?"},{"Start":"06:01.325 ","End":"06:05.795","Text":"Well, that\u0027s the value of x for the middle of the highest category."},{"Start":"06:05.795 ","End":"06:09.260","Text":"If we take a look at this specific histogram,"},{"Start":"06:09.260 ","End":"06:13.250","Text":"that\u0027s the age of Dr. Gold\u0027s patients in years."},{"Start":"06:13.250 ","End":"06:16.690","Text":"The x-axis, that\u0027s age."},{"Start":"06:16.690 ","End":"06:18.890","Text":"We have 3 categories right here,"},{"Start":"06:18.890 ","End":"06:22.130","Text":"from 20-30, 30-40, and 40-60,"},{"Start":"06:22.130 ","End":"06:25.850","Text":"so the mode would be the value of x,"},{"Start":"06:25.850 ","End":"06:29.180","Text":"right here, for the middle of the highest category."},{"Start":"06:29.180 ","End":"06:32.585","Text":"Well, the category having the highest frequency,"},{"Start":"06:32.585 ","End":"06:34.535","Text":"that\u0027s here between 30-40."},{"Start":"06:34.535 ","End":"06:38.470","Text":"The middle of this category right here,"},{"Start":"06:38.470 ","End":"06:40.650","Text":"that\u0027ll be the mode,"},{"Start":"06:40.650 ","End":"06:45.620","Text":"and that\u0027ll be equal to 30 plus 40 divided by 2,"},{"Start":"06:45.620 ","End":"06:47.735","Text":"that equals to 35."},{"Start":"06:47.735 ","End":"06:54.335","Text":"Now, in general, it\u0027s possible for a probability distribution to have more than 1 mode."},{"Start":"06:54.335 ","End":"07:00.625","Text":"For example, let\u0027s say I have another category right here from 60-70,"},{"Start":"07:00.625 ","End":"07:08.120","Text":"and I have the same frequency here as I do for this category right here from 30-40."},{"Start":"07:08.120 ","End":"07:11.150","Text":"That means that I have a second mode."},{"Start":"07:11.150 ","End":"07:15.363","Text":"That\u0027ll be in the middle of this category right here,"},{"Start":"07:15.363 ","End":"07:19.055","Text":"and that\u0027ll be 60 plus 70 divided by 2,"},{"Start":"07:19.055 ","End":"07:21.817","Text":"and that equals to 65,"},{"Start":"07:21.817 ","End":"07:24.485","Text":"so here, I\u0027d have 2 modes,"},{"Start":"07:24.485 ","End":"07:26.615","Text":"one, 35, and one, 65."},{"Start":"07:26.615 ","End":"07:31.985","Text":"Now, another comment. The mode is a relevant value for all types of variables."},{"Start":"07:31.985 ","End":"07:34.189","Text":"Now, remember from the past chapters,"},{"Start":"07:34.189 ","End":"07:37.970","Text":"we studied about the different types of variables, continuous variables,"},{"Start":"07:37.970 ","End":"07:44.850","Text":"discrete variables, so we can apply the mode to any type of variable that we have."}],"ID":13305},{"Watched":false,"Name":"Tutorial 3","Duration":"1m 31s","ChapterTopicVideoID":12836,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12836.jpeg","UploadDate":"2018-07-30T17:30:20.9530000","DurationForVideoObject":"PT1M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.445","Text":"The next metric is called the midrange,"},{"Start":"00:02.445 ","End":"00:06.405","Text":"and that\u0027s denoted by MR for midrange."},{"Start":"00:06.405 ","End":"00:11.580","Text":"The midrange is the average of the highest observation and the lowest observation."},{"Start":"00:11.580 ","End":"00:13.995","Text":"Let\u0027s just take a look at an example here."},{"Start":"00:13.995 ","End":"00:20.745","Text":"Here\u0027s a frequency table of people who own savings plans."},{"Start":"00:20.745 ","End":"00:25.430","Text":"Now, we have here let\u0027s say 100 people who do not own any savings plans,"},{"Start":"00:25.430 ","End":"00:28.205","Text":"75 people who own 1 savings plan,"},{"Start":"00:28.205 ","End":"00:29.750","Text":"25 who own 2,"},{"Start":"00:29.750 ","End":"00:31.540","Text":"and so on and so forth."},{"Start":"00:31.540 ","End":"00:34.250","Text":"How did we calculate the midrange?"},{"Start":"00:34.250 ","End":"00:41.450","Text":"The midrange is the minimum value of x plus the maximum value of x divided by 2."},{"Start":"00:41.450 ","End":"00:46.265","Text":"That\u0027s the average of the highest observation and the lowest observation."},{"Start":"00:46.265 ","End":"00:50.595","Text":"Here\u0027s the x column and here\u0027s the frequency of x."},{"Start":"00:50.595 ","End":"00:52.738","Text":"That\u0027s a frequency column,"},{"Start":"00:52.738 ","End":"00:55.500","Text":"so what\u0027s the minimum value of x right here?"},{"Start":"00:55.500 ","End":"00:59.460","Text":"That\u0027s 0. What\u0027s the maximum value of x?"},{"Start":"00:59.460 ","End":"01:03.495","Text":"That\u0027s 4 divided by 2. That equals to 2."},{"Start":"01:03.495 ","End":"01:09.075","Text":"2 is the value for the midrange of x right here."},{"Start":"01:09.075 ","End":"01:12.935","Text":"Now, on 1 hand this is a very easy metric to calculate."},{"Start":"01:12.935 ","End":"01:15.500","Text":"That\u0027s just the average of 2 numbers."},{"Start":"01:15.500 ","End":"01:20.845","Text":"But on the other hand, is very problematic because it\u0027s based only on these 2 numbers."},{"Start":"01:20.845 ","End":"01:26.818","Text":"It doesn\u0027t take into consideration other values of x and their frequencies,"},{"Start":"01:26.818 ","End":"01:27.920","Text":"so there we have it."},{"Start":"01:27.920 ","End":"01:31.270","Text":"That\u0027s the midrange."}],"ID":13306},{"Watched":false,"Name":"Tutorial 4","Duration":"13m 52s","ChapterTopicVideoID":12837,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12837.jpeg","UploadDate":"2018-07-30T17:31:06.4700000","DurationForVideoObject":"PT13M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"Another important metric is called the median,"},{"Start":"00:02.655 ","End":"00:05.337","Text":"that\u0027s denoted as md."},{"Start":"00:05.337 ","End":"00:08.310","Text":"Now, the median is the value separating"},{"Start":"00:08.310 ","End":"00:11.460","Text":"the higher half of the data from the lower half of the data."},{"Start":"00:11.460 ","End":"00:14.880","Text":"For example, if we have a class full of students and we"},{"Start":"00:14.880 ","End":"00:18.690","Text":"want to know what the median is with respect to height,"},{"Start":"00:18.690 ","End":"00:22.770","Text":"then the median would be the value where half of"},{"Start":"00:22.770 ","End":"00:25.140","Text":"the students would be shorter than that value"},{"Start":"00:25.140 ","End":"00:28.305","Text":"and half the students would be taller than that value."},{"Start":"00:28.305 ","End":"00:34.090","Text":"Let\u0027s see how we can calculate the median in various cases."},{"Start":"00:34.250 ","End":"00:37.945","Text":"Let\u0027s say we get our observation as a list."},{"Start":"00:37.945 ","End":"00:42.890","Text":"Now, what do we need to do in order to calculate the median of that list?"},{"Start":"00:42.890 ","End":"00:45.380","Text":"The first thing that we need to do is we need to"},{"Start":"00:45.380 ","End":"00:48.635","Text":"sort the observations from smallest to largest."},{"Start":"00:48.635 ","End":"00:51.425","Text":"The second thing that we need to do is we need to"},{"Start":"00:51.425 ","End":"00:55.175","Text":"count the number of observations that we have."},{"Start":"00:55.175 ","End":"00:58.610","Text":"Now, if we have an odd number of observations,"},{"Start":"00:58.610 ","End":"01:05.760","Text":"then the median will be the middle observation in the list after we sort the list."},{"Start":"01:05.760 ","End":"01:08.795","Text":"If the number of observations is even,"},{"Start":"01:08.795 ","End":"01:11.780","Text":"then we take the 2 middle observations in"},{"Start":"01:11.780 ","End":"01:15.975","Text":"the list after they\u0027re sorted and we take their average."},{"Start":"01:15.975 ","End":"01:20.155","Text":"Let\u0027s just take a look at an example here."},{"Start":"01:20.155 ","End":"01:24.200","Text":"Let\u0027s say we have a list like this, 7,"},{"Start":"01:24.200 ","End":"01:26.350","Text":"9, 4,"},{"Start":"01:26.350 ","End":"01:29.850","Text":"8, 4, 10, and 6."},{"Start":"01:29.850 ","End":"01:33.635","Text":"The first thing that we want to do is we want to sort this list."},{"Start":"01:33.635 ","End":"01:36.425","Text":"That\u0027ll be 4, 4,"},{"Start":"01:36.425 ","End":"01:38.450","Text":"6, 7,"},{"Start":"01:38.450 ","End":"01:41.155","Text":"8, 9, and 10."},{"Start":"01:41.155 ","End":"01:43.310","Text":"How many observations do we have here?"},{"Start":"01:43.310 ","End":"01:45.530","Text":"Well, we have 1, 2, 3, 4, 5, 6,"},{"Start":"01:45.530 ","End":"01:48.185","Text":"7, so n here equals 7."},{"Start":"01:48.185 ","End":"01:51.815","Text":"We\u0027re dealing with an odd number of observations."},{"Start":"01:51.815 ","End":"01:55.700","Text":"That means our median will be the middle observations."},{"Start":"01:55.700 ","End":"01:59.420","Text":"Now how do we identify this middle observation?"},{"Start":"01:59.420 ","End":"02:06.275","Text":"Well, we calculate the index and this is the index of that middle observation."},{"Start":"02:06.275 ","End":"02:08.240","Text":"If n equals 7,"},{"Start":"02:08.240 ","End":"02:12.050","Text":"then the index of the median would be"},{"Start":"02:12.050 ","End":"02:16.445","Text":"n plus 1 divided by 2 so that\u0027s 7 plus 1 divided by 2,"},{"Start":"02:16.445 ","End":"02:19.805","Text":"that\u0027s 8 divided by 2, and that equals to 4."},{"Start":"02:19.805 ","End":"02:21.725","Text":"Now 4 isn\u0027t the median,"},{"Start":"02:21.725 ","End":"02:26.675","Text":"4 is the index of the observation that will be the median."},{"Start":"02:26.675 ","End":"02:30.835","Text":"We\u0027re looking then for X_4."},{"Start":"02:30.835 ","End":"02:33.765","Text":"This 4 here, that\u0027s this 4 here."},{"Start":"02:33.765 ","End":"02:35.430","Text":"Now, what\u0027s X_4?"},{"Start":"02:35.430 ","End":"02:36.885","Text":"Well, this is X_1,"},{"Start":"02:36.885 ","End":"02:39.540","Text":"X_2, X_3, and X_4."},{"Start":"02:39.540 ","End":"02:44.120","Text":"X_4 equals 7. X_4 will be our median."},{"Start":"02:44.120 ","End":"02:46.835","Text":"Now, let\u0027s take a look at another example."},{"Start":"02:46.835 ","End":"02:50.660","Text":"Let\u0027s say we have a list, which is 8,"},{"Start":"02:50.660 ","End":"02:52.685","Text":"10, 9,"},{"Start":"02:52.685 ","End":"02:56.160","Text":"5, 4, and 7."},{"Start":"02:56.270 ","End":"02:59.090","Text":"The first thing we need to do is we need to"},{"Start":"02:59.090 ","End":"03:02.240","Text":"sort the observations from smallest to largest."},{"Start":"03:02.240 ","End":"03:04.475","Text":"That\u0027ll be 4, 5,"},{"Start":"03:04.475 ","End":"03:06.325","Text":"7, 8,"},{"Start":"03:06.325 ","End":"03:07.835","Text":"9, and 10."},{"Start":"03:07.835 ","End":"03:10.085","Text":"How many observations do we have?"},{"Start":"03:10.085 ","End":"03:11.690","Text":"1, 2, 3, 4, 5,"},{"Start":"03:11.690 ","End":"03:14.300","Text":"6, so n here equals to 6."},{"Start":"03:14.300 ","End":"03:18.290","Text":"That means that we have an even number of observations."},{"Start":"03:18.290 ","End":"03:23.690","Text":"What we need to do is we need to take the middle 2 observations,"},{"Start":"03:23.690 ","End":"03:26.555","Text":"take their average and that would be the median."},{"Start":"03:26.555 ","End":"03:30.860","Text":"Now, how do we identify the middle 2 observations?"},{"Start":"03:30.860 ","End":"03:34.020","Text":"Well, that\u0027s their indices right here."},{"Start":"03:34.020 ","End":"03:36.500","Text":"The first index is n divided by 2,"},{"Start":"03:36.500 ","End":"03:38.120","Text":"so that\u0027ll be 6 divided by 2,"},{"Start":"03:38.120 ","End":"03:39.740","Text":"that equals to 3,"},{"Start":"03:39.740 ","End":"03:41.840","Text":"and n divided by 2 plus 1,"},{"Start":"03:41.840 ","End":"03:43.535","Text":"that\u0027ll be 3 plus 1,"},{"Start":"03:43.535 ","End":"03:45.565","Text":"that equals to 4."},{"Start":"03:45.565 ","End":"03:54.840","Text":"We\u0027re looking for X_3 and X_4."},{"Start":"03:54.840 ","End":"03:56.535","Text":"Now, X_3, that\u0027s X_1, X_2,"},{"Start":"03:56.535 ","End":"03:58.485","Text":"X_3 that equals to 7,"},{"Start":"03:58.485 ","End":"04:01.290","Text":"and X_4, that equals to 8."},{"Start":"04:01.290 ","End":"04:06.140","Text":"Now, how do we calculate the median?"},{"Start":"04:06.140 ","End":"04:09.350","Text":"We take the average of these numbers."},{"Start":"04:09.350 ","End":"04:11.900","Text":"That\u0027s what this expression means."},{"Start":"04:11.900 ","End":"04:17.660","Text":"The median is 7 plus 8 divided by 2,"},{"Start":"04:17.660 ","End":"04:20.540","Text":"and that equals to 15 divided by 2,"},{"Start":"04:20.540 ","End":"04:22.430","Text":"and that equals to 7 and a half."},{"Start":"04:22.430 ","End":"04:24.470","Text":"First of all, this is the median."},{"Start":"04:24.470 ","End":"04:25.850","Text":"We\u0027ve calculated that."},{"Start":"04:25.850 ","End":"04:29.180","Text":"That\u0027s the value where half of"},{"Start":"04:29.180 ","End":"04:34.235","Text":"the observations are below this value and half of the observations are above this value."},{"Start":"04:34.235 ","End":"04:36.365","Text":"Now, we have to pay attention here,"},{"Start":"04:36.365 ","End":"04:38.050","Text":"7 and a half,"},{"Start":"04:38.050 ","End":"04:40.340","Text":"this is a value of the median."},{"Start":"04:40.340 ","End":"04:43.655","Text":"We don\u0027t see this value in the observations,"},{"Start":"04:43.655 ","End":"04:47.210","Text":"and it\u0027s not necessary for us to do that."},{"Start":"04:47.210 ","End":"04:52.580","Text":"The median doesn\u0027t have to be 1 of the values of the observations."},{"Start":"04:52.580 ","End":"04:56.720","Text":"It has to be some point in the list where half of"},{"Start":"04:56.720 ","End":"05:03.280","Text":"the observations are below this value and half of the observations are above this value."},{"Start":"05:03.280 ","End":"05:06.800","Text":"How do we calculate the median when the data is"},{"Start":"05:06.800 ","End":"05:10.070","Text":"presented to us in a discrete frequency table?"},{"Start":"05:10.070 ","End":"05:13.925","Text":"We use the same process that we described in the list above."},{"Start":"05:13.925 ","End":"05:16.960","Text":"But here we use the cumulative frequency."},{"Start":"05:16.960 ","End":"05:20.070","Text":"That\u0027ll help us to calculate a median."},{"Start":"05:20.070 ","End":"05:24.320","Text":"We\u0027ll use first the data that was"},{"Start":"05:24.320 ","End":"05:28.265","Text":"given to us about the distribution of people who have saving plans."},{"Start":"05:28.265 ","End":"05:30.860","Text":"Here we have 100 people who have no saving plans,"},{"Start":"05:30.860 ","End":"05:32.840","Text":"75 people who have 1 saving plan,"},{"Start":"05:32.840 ","End":"05:34.055","Text":"and so on and so forth."},{"Start":"05:34.055 ","End":"05:36.380","Text":"How many people in total do we have here?"},{"Start":"05:36.380 ","End":"05:40.880","Text":"We sum up these guys right here and that comes out to 250."},{"Start":"05:40.880 ","End":"05:45.380","Text":"Now let\u0027s just calculate the cumulative frequency here."},{"Start":"05:45.380 ","End":"05:49.660","Text":"That\u0027ll be 100, that\u0027s 175,"},{"Start":"05:49.660 ","End":"05:57.730","Text":"that\u0027s 200, 225, and 250."},{"Start":"05:58.670 ","End":"06:03.945","Text":"Because n here, that\u0027s an even number,"},{"Start":"06:03.945 ","End":"06:08.900","Text":"then the equation for the median would be like this."},{"Start":"06:08.900 ","End":"06:19.460","Text":"That\u0027ll be X_n divided by 2 plus X_n divided by 2 plus 1 divided by 2."},{"Start":"06:19.460 ","End":"06:23.500","Text":"Now, what\u0027s n divided by 2?"},{"Start":"06:23.500 ","End":"06:29.625","Text":"N divided by 2 is 250 divided by 2 that equals to 125."},{"Start":"06:29.625 ","End":"06:31.790","Text":"What\u0027s n divided by 2 plus 1?"},{"Start":"06:31.790 ","End":"06:35.660","Text":"That would be 250 divided by 2 plus 1,"},{"Start":"06:35.660 ","End":"06:38.900","Text":"and that equals to 126."},{"Start":"06:38.900 ","End":"06:43.460","Text":"Now, we don\u0027t take the average of these numbers right here."},{"Start":"06:43.460 ","End":"06:46.355","Text":"These numbers are just placeholders."},{"Start":"06:46.355 ","End":"06:50.030","Text":"What we need to do is we need to calculate the value"},{"Start":"06:50.030 ","End":"06:57.205","Text":"of X_125 and X_126."},{"Start":"06:57.205 ","End":"07:00.875","Text":"Well, what are these values right here then?"},{"Start":"07:00.875 ","End":"07:08.910","Text":"Assume that all 250 people are standing in line."},{"Start":"07:08.910 ","End":"07:12.730","Text":"Now what we do is we take the first 100 people,"},{"Start":"07:12.730 ","End":"07:19.040","Text":"we give them the value of 0 because that\u0027s the number of saving plans that they have."},{"Start":"07:19.040 ","End":"07:22.190","Text":"Now the next 75 people,"},{"Start":"07:22.190 ","End":"07:25.375","Text":"they have 1 savings plan."},{"Start":"07:25.375 ","End":"07:28.439","Text":"The next 25 people have 2 saving plan,"},{"Start":"07:28.439 ","End":"07:29.520","Text":"and so on and so forth."},{"Start":"07:29.520 ","End":"07:34.700","Text":"We\u0027re looking for the person in place number 125."},{"Start":"07:34.700 ","End":"07:36.940","Text":"He\u0027s not in this category,"},{"Start":"07:36.940 ","End":"07:40.360","Text":"he\u0027s obviously in this category right here."},{"Start":"07:40.360 ","End":"07:45.915","Text":"That means that this guy has 1 saving plan."},{"Start":"07:45.915 ","End":"07:51.025","Text":"Same thing for person in place number 126."},{"Start":"07:51.025 ","End":"07:55.979","Text":"He also has 1 saving plan and person number 127"},{"Start":"07:55.979 ","End":"08:01.830","Text":"and 128 and so on and so forth until place number 175."},{"Start":"08:02.870 ","End":"08:09.090","Text":"Here, these are the values for X_125 and X_126."},{"Start":"08:09.090 ","End":"08:14.240","Text":"That equals to 1 plus 1 divided by 2 and that equals to 1."},{"Start":"08:14.240 ","End":"08:19.635","Text":"This then, that\u0027s the median of the data,"},{"Start":"08:19.635 ","End":"08:23.375","Text":"when the data is presented to us in a discrete frequency table."},{"Start":"08:23.375 ","End":"08:25.490","Text":"Now, if we have a bar chart,"},{"Start":"08:25.490 ","End":"08:27.815","Text":"the data is presented to us in a bar chart,"},{"Start":"08:27.815 ","End":"08:31.080","Text":"it\u0027s very difficult for us to calculate the median."},{"Start":"08:31.080 ","End":"08:33.700","Text":"What we do is we convert the graph to"},{"Start":"08:33.700 ","End":"08:37.260","Text":"a discrete frequency table in order to find the median."},{"Start":"08:37.260 ","End":"08:39.670","Text":"Just like we did right here."},{"Start":"08:40.090 ","End":"08:44.930","Text":"When we get data in a table of frequencies by categories,"},{"Start":"08:44.930 ","End":"08:46.955","Text":"then we use the following technique."},{"Start":"08:46.955 ","End":"08:53.780","Text":"Now, the first step is to find the median category whose location is n divided by 2."},{"Start":"08:53.780 ","End":"08:55.235","Text":"Then the second stage,"},{"Start":"08:55.235 ","End":"08:59.950","Text":"we use this formula to calculate the median."},{"Start":"08:59.950 ","End":"09:01.910","Text":"Let\u0027s just see what we have here."},{"Start":"09:01.910 ","End":"09:07.265","Text":"Here we have a distribution of students receiving grades."},{"Start":"09:07.265 ","End":"09:10.160","Text":"X is the grades and f of x,"},{"Start":"09:10.160 ","End":"09:15.545","Text":"that\u0027s the number of students receiving grades in these categories."},{"Start":"09:15.545 ","End":"09:18.260","Text":"We have here the first category from 0-60,"},{"Start":"09:18.260 ","End":"09:20.960","Text":"second category from 60-70,"},{"Start":"09:20.960 ","End":"09:22.145","Text":"and so on and so forth."},{"Start":"09:22.145 ","End":"09:28.670","Text":"We have here 20 students who received grades in this range right here from 0-60,"},{"Start":"09:28.670 ","End":"09:33.020","Text":"10 students who received grades from 60-70,"},{"Start":"09:33.020 ","End":"09:34.700","Text":"18 from 70-80,"},{"Start":"09:34.700 ","End":"09:36.245","Text":"and so on and so forth."},{"Start":"09:36.245 ","End":"09:39.900","Text":"Now, how many students in total do we have here?"},{"Start":"09:39.900 ","End":"09:43.300","Text":"Well, n equals to 78."},{"Start":"09:43.880 ","End":"09:50.327","Text":"Let\u0027s just calculate or identify the median category right here."},{"Start":"09:50.327 ","End":"09:53.895","Text":"That equals, n divide by 2."},{"Start":"09:53.895 ","End":"09:56.520","Text":"That\u0027s 78 divided by 2,"},{"Start":"09:56.520 ","End":"09:58.035","Text":"that equals to 39."},{"Start":"09:58.035 ","End":"10:03.208","Text":"We\u0027re looking for the observation in the 39th place."},{"Start":"10:03.208 ","End":"10:08.800","Text":"We need to identify the category that holds this observation."},{"Start":"10:08.800 ","End":"10:11.105","Text":"In order to do that,"},{"Start":"10:11.105 ","End":"10:14.825","Text":"let\u0027s just calculate the cumulative frequency here."},{"Start":"10:14.825 ","End":"10:18.110","Text":"That\u0027ll be F of x."},{"Start":"10:18.110 ","End":"10:21.690","Text":"That\u0027ll be 20, 30,"},{"Start":"10:21.690 ","End":"10:28.380","Text":"48, 63, and 78."},{"Start":"10:28.380 ","End":"10:36.380","Text":"What category holds the observation in the 39th place or the 39 observation."},{"Start":"10:36.380 ","End":"10:40.160","Text":"It\u0027s not the first category and it\u0027s not the second category."},{"Start":"10:40.160 ","End":"10:44.915","Text":"It\u0027s the third category right here. Why is that?"},{"Start":"10:44.915 ","End":"10:49.890","Text":"The first category only holds the first 20 students."},{"Start":"10:52.040 ","End":"10:56.790","Text":"The second category holds the next 10 students."},{"Start":"10:56.790 ","End":"11:02.190","Text":"Only 30 students in total and we want student number 39 here."},{"Start":"11:02.190 ","End":"11:05.019","Text":"That\u0027s in the third category."},{"Start":"11:05.060 ","End":"11:11.540","Text":"The next thing that we do is we use this formula to calculate the median."},{"Start":"11:11.540 ","End":"11:13.445","Text":"Now, this looks like a scary formula,"},{"Start":"11:13.445 ","End":"11:15.005","Text":"but it really isn\u0027t."},{"Start":"11:15.005 ","End":"11:17.630","Text":"On 1 hand, and on the other hand,"},{"Start":"11:17.630 ","End":"11:20.510","Text":"you\u0027ll have enough time to do the exercises and"},{"Start":"11:20.510 ","End":"11:24.050","Text":"get comfortable with this. Let\u0027s get to work."},{"Start":"11:24.050 ","End":"11:28.090","Text":"The median is equal to L_0."},{"Start":"11:28.090 ","End":"11:32.600","Text":"What\u0027s L_0? That\u0027s the lower limit of the category,"},{"Start":"11:32.600 ","End":"11:35.250","Text":"that\u0027s 70 right here."},{"Start":"11:36.020 ","End":"11:38.400","Text":"Plus, now,"},{"Start":"11:38.400 ","End":"11:41.175","Text":"n divided by 2 that\u0027s right here,"},{"Start":"11:41.175 ","End":"11:45.465","Text":"39 minus, now,"},{"Start":"11:45.465 ","End":"11:49.160","Text":"F of X_m minus 1."},{"Start":"11:49.160 ","End":"11:50.660","Text":"This expression,"},{"Start":"11:50.660 ","End":"11:56.030","Text":"that\u0027s a cumulative frequency of the category proceeding the median category."},{"Start":"11:56.030 ","End":"11:59.360","Text":"That means that if this is the medium category,"},{"Start":"11:59.360 ","End":"12:04.562","Text":"then the cumulative frequency of the category before that, that\u0027s 30."},{"Start":"12:04.562 ","End":"12:09.515","Text":"That\u0027ll be 30 divided by,"},{"Start":"12:09.515 ","End":"12:11.810","Text":"f of X_m,"},{"Start":"12:11.810 ","End":"12:15.170","Text":"that equals to the frequency of the medium category."},{"Start":"12:15.170 ","End":"12:16.865","Text":"Here\u0027s the median category again,"},{"Start":"12:16.865 ","End":"12:25.890","Text":"and this is a frequency that\u0027s 18 times L_1 minus L_0."},{"Start":"12:25.890 ","End":"12:29.960","Text":"L_1 is the upper limit of the category that\u0027s right here, that\u0027s 80,"},{"Start":"12:29.960 ","End":"12:35.945","Text":"minus the lower limit of the category, 70."},{"Start":"12:35.945 ","End":"12:38.600","Text":"Well, after doing all the math right here,"},{"Start":"12:38.600 ","End":"12:40.550","Text":"that comes out to 75,"},{"Start":"12:40.550 ","End":"12:42.560","Text":"so 75 is the median."},{"Start":"12:42.560 ","End":"12:46.460","Text":"That means that half of the students received grades below"},{"Start":"12:46.460 ","End":"12:52.080","Text":"75 and half of the students received grades above 75."},{"Start":"12:52.580 ","End":"12:56.050","Text":"If the data is presented to us in a histogram,"},{"Start":"12:56.050 ","End":"12:59.590","Text":"then I suggest that we transform the data into"},{"Start":"12:59.590 ","End":"13:03.775","Text":"a frequency table by categories and then calculate the median."},{"Start":"13:03.775 ","End":"13:06.670","Text":"But if we have to deal with the histogram,"},{"Start":"13:06.670 ","End":"13:09.520","Text":"then the median is the value of"},{"Start":"13:09.520 ","End":"13:13.720","Text":"the x-axis that divides the histogram into 2 parts with equal areas."},{"Start":"13:13.720 ","End":"13:17.770","Text":"That means that if this is our histogram right here and that\u0027s our x-axis,"},{"Start":"13:17.770 ","End":"13:19.930","Text":"we have to find this value right"},{"Start":"13:19.930 ","End":"13:25.390","Text":"here that divides the histogram into 2 with respect to area."},{"Start":"13:25.390 ","End":"13:29.530","Text":"That means that this area right here to the left of"},{"Start":"13:29.530 ","End":"13:35.515","Text":"this value has to be equal to this area right here to the right of this value."},{"Start":"13:35.515 ","End":"13:40.940","Text":"If we find this value that separates the area into 2 equal parts,"},{"Start":"13:40.940 ","End":"13:43.280","Text":"then this value would be the median."},{"Start":"13:43.280 ","End":"13:45.185","Text":"Now, as a general comment,"},{"Start":"13:45.185 ","End":"13:52.320","Text":"the median is irrelevant when dealing with a nominal variable or a qualitative variable."}],"ID":13307},{"Watched":false,"Name":"Tutorial 5","Duration":"4m 49s","ChapterTopicVideoID":12838,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12838.jpeg","UploadDate":"2018-07-30T17:31:21.1370000","DurationForVideoObject":"PT4M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.265","Text":"The next metric is the average."},{"Start":"00:02.265 ","End":"00:08.580","Text":"This is the most common metric that\u0027s used to measure the center of the data."},{"Start":"00:08.580 ","End":"00:16.905","Text":"We can look at it as the center of gravity of all the observations in our dataset."},{"Start":"00:16.905 ","End":"00:20.070","Text":"If the data is presented to us in a list,"},{"Start":"00:20.070 ","End":"00:23.595","Text":"all we have to do is we have to add"},{"Start":"00:23.595 ","End":"00:31.095","Text":"all the values of the observations and divide it by the number of observations."},{"Start":"00:31.095 ","End":"00:32.685","Text":"That\u0027ll be our average."},{"Start":"00:32.685 ","End":"00:36.025","Text":"The average is denoted by x bar, like this."},{"Start":"00:36.025 ","End":"00:39.590","Text":"Again, that equals to the sum of all x divided by"},{"Start":"00:39.590 ","End":"00:43.570","Text":"n. How many observations do we have here?"},{"Start":"00:43.570 ","End":"00:46.084","Text":"N here equals to 7."},{"Start":"00:46.084 ","End":"00:49.760","Text":"What we need to do is we need to add up all the values."},{"Start":"00:49.760 ","End":"00:56.970","Text":"That\u0027ll be 7 plus 9 plus 4 plus 8 plus 4 plus 10 plus 6,"},{"Start":"00:56.970 ","End":"01:00.000","Text":"and we have to divide that by 7."},{"Start":"01:00.000 ","End":"01:05.590","Text":"That\u0027ll come out to 6.857."},{"Start":"01:06.430 ","End":"01:09.850","Text":"If the data is given to us in a frequency table,"},{"Start":"01:09.850 ","End":"01:13.760","Text":"then the average would be the sum of x times"},{"Start":"01:13.760 ","End":"01:18.200","Text":"the frequency of x divided by the number of observations."},{"Start":"01:18.200 ","End":"01:22.130","Text":"Let\u0027s just take a look at an example here we have the distribution"},{"Start":"01:22.130 ","End":"01:25.940","Text":"of people who have saving plans."},{"Start":"01:25.940 ","End":"01:29.630","Text":"We have here, let\u0027s say 100 people who have no savings plan,"},{"Start":"01:29.630 ","End":"01:31.895","Text":"75 who have 1 saving plans,"},{"Start":"01:31.895 ","End":"01:35.075","Text":"25 people have 2 saving plans and so on and so forth."},{"Start":"01:35.075 ","End":"01:40.115","Text":"We want to know what\u0027s the average number of saving plans per person."},{"Start":"01:40.115 ","End":"01:42.710","Text":"The first thing that we want to"},{"Start":"01:42.710 ","End":"01:45.350","Text":"do is we want to find out how many people that we have now"},{"Start":"01:45.350 ","End":"01:52.385","Text":"that means n. N is just the sum of the frequencies here and that\u0027ll be 250."},{"Start":"01:52.385 ","End":"01:57.380","Text":"We can\u0027t just take the average of these numbers right here"},{"Start":"01:57.380 ","End":"02:03.245","Text":"because each number represents a person with a different number of saving plans."},{"Start":"02:03.245 ","End":"02:05.885","Text":"What we need to do is this."},{"Start":"02:05.885 ","End":"02:09.500","Text":"We have to sum x,"},{"Start":"02:09.500 ","End":"02:11.870","Text":"the number of saving plans times the frequencies."},{"Start":"02:11.870 ","End":"02:21.785","Text":"So here that\u0027ll be 0 times 100 plus 1 times 75 plus 2 times 25,"},{"Start":"02:21.785 ","End":"02:27.995","Text":"plus 3 times 25 plus 4 times 25."},{"Start":"02:27.995 ","End":"02:32.705","Text":"All this divided by 250."},{"Start":"02:32.705 ","End":"02:37.220","Text":"That comes out to 1.2,"},{"Start":"02:37.220 ","End":"02:40.130","Text":"that\u0027s our average, that\u0027s our x bar."},{"Start":"02:40.130 ","End":"02:50.639","Text":"All that means is that we have an average of 1.2 saving plans per person in our dataset."},{"Start":"02:51.280 ","End":"02:56.570","Text":"When the data is presented to us in a frequency table with categories,"},{"Start":"02:56.570 ","End":"02:59.360","Text":"then we use the same formula as we did above,"},{"Start":"02:59.360 ","End":"03:03.635","Text":"but we use the middle of the category for the x value."},{"Start":"03:03.635 ","End":"03:06.800","Text":"This average is an approximation obviously."},{"Start":"03:06.800 ","End":"03:08.675","Text":"What do we mean by that?"},{"Start":"03:08.675 ","End":"03:13.850","Text":"Well, we say that x bar equals the sum of x times"},{"Start":"03:13.850 ","End":"03:19.250","Text":"the frequency of x divided by n. What\u0027s our n right here?"},{"Start":"03:19.250 ","End":"03:22.370","Text":"Well, that equals to 78."},{"Start":"03:22.370 ","End":"03:26.900","Text":"What\u0027s the middle of these categories right here?"},{"Start":"03:26.900 ","End":"03:29.285","Text":"What\u0027s our midpoint right here?"},{"Start":"03:29.285 ","End":"03:34.475","Text":"Well, that\u0027s the upper bound minus the lower bound divided by 2."},{"Start":"03:34.475 ","End":"03:38.960","Text":"That\u0027ll be 16 minus 0 divided by 2, that\u0027ll be 30."},{"Start":"03:38.960 ","End":"03:41.390","Text":"Here that\u0027ll be 65,"},{"Start":"03:41.390 ","End":"03:43.690","Text":"here that\u0027ll be 75,"},{"Start":"03:43.690 ","End":"03:46.050","Text":"here that\u0027ll be 85,"},{"Start":"03:46.050 ","End":"03:48.615","Text":"and here that\u0027ll be 95."},{"Start":"03:48.615 ","End":"03:50.810","Text":"Now that we have the middle points,"},{"Start":"03:50.810 ","End":"03:53.435","Text":"we can calculate the average."},{"Start":"03:53.435 ","End":"03:56.460","Text":"Because then the average would be 30,"},{"Start":"03:56.460 ","End":"03:59.045","Text":"that\u0027s the midpoint of the category,"},{"Start":"03:59.045 ","End":"04:00.290","Text":"times the frequency,"},{"Start":"04:00.290 ","End":"04:02.255","Text":"where the frequency is 20,"},{"Start":"04:02.255 ","End":"04:11.825","Text":"plus 65 times 10 plus 75 times 18 plus 85 times 15,"},{"Start":"04:11.825 ","End":"04:15.530","Text":"plus 95 times 15."},{"Start":"04:15.530 ","End":"04:21.140","Text":"All that has to be divided by 78,"},{"Start":"04:21.140 ","End":"04:28.485","Text":"so that equals to 67.95."},{"Start":"04:28.485 ","End":"04:38.245","Text":"All this means is that our average grade of students is 67.95."},{"Start":"04:38.245 ","End":"04:40.415","Text":"Just a general comment."},{"Start":"04:40.415 ","End":"04:43.220","Text":"The average is relevant only when dealing with"},{"Start":"04:43.220 ","End":"04:48.780","Text":"quantitative variables not with qualitative or nominal values."}],"ID":13308},{"Watched":false,"Name":"Tutorial 6","Duration":"3m 33s","ChapterTopicVideoID":12839,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12839.jpeg","UploadDate":"2018-07-30T17:31:31.3170000","DurationForVideoObject":"PT3M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"Here we\u0027ll talk about the central location measures in special probability distributions."},{"Start":"00:05.085 ","End":"00:06.735","Text":"Now in previous chapters,"},{"Start":"00:06.735 ","End":"00:09.300","Text":"we talked about the different shapes of distributions,"},{"Start":"00:09.300 ","End":"00:14.145","Text":"specifically the symmetrical distributions that are bell shaped."},{"Start":"00:14.145 ","End":"00:19.637","Text":"Now here we see a distribution that\u0027s bell-shaped and it\u0027s symmetrical."},{"Start":"00:19.637 ","End":"00:21.525","Text":"Now, in this type of distribution,"},{"Start":"00:21.525 ","End":"00:24.600","Text":"all the measures of central tendency are equal to"},{"Start":"00:24.600 ","End":"00:27.945","Text":"each other and that\u0027s because of the symmetry."},{"Start":"00:27.945 ","End":"00:33.435","Text":"Now, not all symmetrical distributions are bell-shaped."},{"Start":"00:33.435 ","End":"00:37.560","Text":"Here we have a symmetrical distribution but it\u0027s U shaped,"},{"Start":"00:37.560 ","End":"00:38.745","Text":"it\u0027s not bell-shaped,"},{"Start":"00:38.745 ","End":"00:40.500","Text":"and as such,"},{"Start":"00:40.500 ","End":"00:47.090","Text":"we don\u0027t have all the measures of central tendency being equal to each other."},{"Start":"00:47.090 ","End":"00:50.540","Text":"We have the average and the median equaling each other,"},{"Start":"00:50.540 ","End":"00:56.660","Text":"and that\u0027s in the middle, but the mode is not equal to the average and median."},{"Start":"00:56.660 ","End":"01:02.434","Text":"Why is that? Well, the mode is defined as the point where we have the highest frequency."},{"Start":"01:02.434 ","End":"01:04.040","Text":"Now in a U-shape distribution,"},{"Start":"01:04.040 ","End":"01:06.330","Text":"we have 2 such points,"},{"Start":"01:07.090 ","End":"01:09.830","Text":"and those points are at the sides of the distribution,"},{"Start":"01:09.830 ","End":"01:12.843","Text":"we have that right here and right here,"},{"Start":"01:12.843 ","End":"01:20.105","Text":"so we have here 2 modes and the average and median equaling each other in the middle."},{"Start":"01:20.105 ","End":"01:24.357","Text":"Distributions can also be asymmetric in nature."},{"Start":"01:24.357 ","End":"01:28.940","Text":"Here we have a right or positive asymmetric probability distribution,"},{"Start":"01:28.940 ","End":"01:34.025","Text":"and here we have the left or negative asymmetric probability distribution."},{"Start":"01:34.025 ","End":"01:37.080","Text":"Now I\u0027ll be talking about 1 of"},{"Start":"01:37.080 ","End":"01:41.144","Text":"the shapes because 1 shape is just the mirror image of the other,"},{"Start":"01:41.144 ","End":"01:43.840","Text":"so let\u0027s talk about this shape right here,"},{"Start":"01:43.840 ","End":"01:46.769","Text":"the right asymmetric distribution."},{"Start":"01:46.769 ","End":"01:49.370","Text":"What happens to these metrics,"},{"Start":"01:49.370 ","End":"01:52.145","Text":"these measures of central tendency?"},{"Start":"01:52.145 ","End":"01:58.350","Text":"The mode here, that\u0027s the x value that has the highest frequency,"},{"Start":"01:58.350 ","End":"02:04.490","Text":"and since we have a right or positive asymmetric shape to this distribution,"},{"Start":"02:04.490 ","End":"02:11.180","Text":"then the highest frequency is the left most value of x right here."},{"Start":"02:11.180 ","End":"02:13.100","Text":"What about the median?"},{"Start":"02:13.100 ","End":"02:16.340","Text":"Well, the median is pulled to the right. Why is that?"},{"Start":"02:16.340 ","End":"02:20.180","Text":"Well, the median has to catch a 50 percent of"},{"Start":"02:20.180 ","End":"02:24.589","Text":"the area under the curve to the right and 50 percent to the left."},{"Start":"02:24.589 ","End":"02:27.890","Text":"That means that the median has to be greater than the mode."},{"Start":"02:27.890 ","End":"02:29.525","Text":"Now, what about the average?"},{"Start":"02:29.525 ","End":"02:32.450","Text":"Well, the average is influenced by higher value,"},{"Start":"02:32.450 ","End":"02:36.062","Text":"so that\u0027s pulled also to the right."},{"Start":"02:36.062 ","End":"02:38.634","Text":"What about the mid range?"},{"Start":"02:38.634 ","End":"02:43.220","Text":"Well, this metric is influenced by only 2 values,"},{"Start":"02:43.220 ","End":"02:46.925","Text":"the x minimum and x maximum."},{"Start":"02:46.925 ","End":"02:50.990","Text":"Now, since this distribution is asymmetric to"},{"Start":"02:50.990 ","End":"02:55.655","Text":"the right with the maximum value of x as a very high number,"},{"Start":"02:55.655 ","End":"03:01.075","Text":"that means that the mid-range is being pulled as well to the right."},{"Start":"03:01.075 ","End":"03:06.330","Text":"This is the order of a right positive asymmetric distribution,"},{"Start":"03:06.330 ","End":"03:09.650","Text":"this is the order of the central measure of tendency,"},{"Start":"03:09.650 ","End":"03:12.395","Text":"we have the mode, then the median,"},{"Start":"03:12.395 ","End":"03:15.335","Text":"then the average, then the mid range."},{"Start":"03:15.335 ","End":"03:21.110","Text":"As we said, if we have a left or negative asymmetric distribution,"},{"Start":"03:21.110 ","End":"03:23.720","Text":"this is the mirror image,"},{"Start":"03:23.720 ","End":"03:29.600","Text":"so the mode will be the rightmost value and then the median,"},{"Start":"03:29.600 ","End":"03:30.785","Text":"and then the average,"},{"Start":"03:30.785 ","End":"03:33.210","Text":"and then the mid range."}],"ID":13309},{"Watched":false,"Name":"Exercise 1","Duration":"4m 1s","ChapterTopicVideoID":12826,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12826.jpeg","UploadDate":"2018-07-30T17:26:50.3070000","DurationForVideoObject":"PT4M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this question, we are given a list of marks of"},{"Start":"00:02.430 ","End":"00:05.040","Text":"20 students on a reading comprehension test."},{"Start":"00:05.040 ","End":"00:07.860","Text":"These are the marks right here and we\u0027re asked to calculate the median,"},{"Start":"00:07.860 ","End":"00:10.500","Text":"the mode, and the average of the marks."},{"Start":"00:10.500 ","End":"00:14.444","Text":"First thing that we need to do is to identify a variable."},{"Start":"00:14.444 ","End":"00:21.260","Text":"Now, our variable, that\u0027ll be the marks."},{"Start":"00:21.260 ","End":"00:23.670","Text":"This is what we\u0027re looking at."},{"Start":"00:23.670 ","End":"00:30.105","Text":"This is a quantitative variable and it\u0027s a quantitative discrete variable."},{"Start":"00:30.105 ","End":"00:34.485","Text":"The number of observations that we have here is 20,"},{"Start":"00:34.485 ","End":"00:37.070","Text":"that\u0027s given to us right here."},{"Start":"00:37.070 ","End":"00:41.300","Text":"Let\u0027s first calculate the mode."},{"Start":"00:41.300 ","End":"00:48.950","Text":"What\u0027s the mode? Mode is defined as the value that has the highest frequency."},{"Start":"00:48.950 ","End":"00:54.260","Text":"Let\u0027s just draw a quick frequency table to calculate the mode."},{"Start":"00:54.260 ","End":"00:59.520","Text":"The minimum value here is 4 and the maximum is 10."},{"Start":"00:59.520 ","End":"01:02.120","Text":"Let\u0027s just do that. That\u0027ll be our x values,"},{"Start":"01:02.120 ","End":"01:04.765","Text":"that will be our frequency."},{"Start":"01:04.765 ","End":"01:08.490","Text":"Okay? We have here 4, 5,"},{"Start":"01:08.490 ","End":"01:10.035","Text":"6, 7,"},{"Start":"01:10.035 ","End":"01:12.975","Text":"8, 9 and 10."},{"Start":"01:12.975 ","End":"01:15.210","Text":"How many fours do I have here?"},{"Start":"01:15.210 ","End":"01:18.330","Text":"I have 1 4. How many fives?"},{"Start":"01:18.330 ","End":"01:21.390","Text":"That\u0027s 1 and 2."},{"Start":"01:21.390 ","End":"01:23.325","Text":"How many sixes?"},{"Start":"01:23.325 ","End":"01:25.305","Text":"1, 2, 3,"},{"Start":"01:25.305 ","End":"01:27.180","Text":"4, 5, and 6."},{"Start":"01:27.180 ","End":"01:28.530","Text":"I have 6 sixes."},{"Start":"01:28.530 ","End":"01:29.880","Text":"How many sevens?"},{"Start":"01:29.880 ","End":"01:31.320","Text":"1, 2,"},{"Start":"01:31.320 ","End":"01:33.210","Text":"3, and 4."},{"Start":"01:33.210 ","End":"01:35.355","Text":"How many eights? 1,"},{"Start":"01:35.355 ","End":"01:37.800","Text":"2, 3, and 4."},{"Start":"01:37.800 ","End":"01:39.120","Text":"How many nines?"},{"Start":"01:39.120 ","End":"01:43.360","Text":"We have 2 nines and 1 10."},{"Start":"01:43.670 ","End":"01:50.090","Text":"As we said, the mode is the x value that has the highest frequency."},{"Start":"01:50.090 ","End":"01:52.655","Text":"The highest frequency is right here."},{"Start":"01:52.655 ","End":"01:55.445","Text":"The x value that has the highest frequency,"},{"Start":"01:55.445 ","End":"01:57.925","Text":"that\u0027s this guy right here, that\u0027s 6."},{"Start":"01:57.925 ","End":"02:00.315","Text":"6 would be the mode."},{"Start":"02:00.315 ","End":"02:03.190","Text":"What about the median?"},{"Start":"02:04.130 ","End":"02:07.945","Text":"We have an even number of observations."},{"Start":"02:07.945 ","End":"02:14.960","Text":"The calculation of the median is the average of the 2 middle observations."},{"Start":"02:14.960 ","End":"02:19.545","Text":"That means that I have x sub n divided by 2,"},{"Start":"02:19.545 ","End":"02:22.200","Text":"plus x sub n divided by 2,"},{"Start":"02:22.200 ","End":"02:25.860","Text":"plus 1 divided by 2."},{"Start":"02:25.860 ","End":"02:28.115","Text":"In order to calculate that,"},{"Start":"02:28.115 ","End":"02:33.150","Text":"I need to order these observations. Let\u0027s do that."},{"Start":"02:33.370 ","End":"02:38.315","Text":"Here are the observations after I ordered them."},{"Start":"02:38.315 ","End":"02:47.040","Text":"We\u0027re looking for x sub n divided by 2. n divided by 2,"},{"Start":"02:47.040 ","End":"02:50.110","Text":"that equals to 20 divided by 2,"},{"Start":"02:50.110 ","End":"02:51.580","Text":"that equals to 10."},{"Start":"02:51.580 ","End":"02:57.940","Text":"So we\u0027re looking at x sub 10 plus x sub 11 divided by 2."},{"Start":"02:57.940 ","End":"03:00.745","Text":"What\u0027s the value of x sub 10?"},{"Start":"03:00.745 ","End":"03:05.280","Text":"That\u0027s the 10th x in the ordered list."},{"Start":"03:05.280 ","End":"03:07.200","Text":"Let\u0027s take a look at what that is."},{"Start":"03:07.200 ","End":"03:08.910","Text":"That\u0027s 1, 2, 3, 4, 5,"},{"Start":"03:08.910 ","End":"03:11.055","Text":"6, 7, 8, 9, 10."},{"Start":"03:11.055 ","End":"03:15.510","Text":"That\u0027s a 7. What about the 11th x?"},{"Start":"03:15.510 ","End":"03:16.950","Text":"That\u0027s also a 7."},{"Start":"03:16.950 ","End":"03:21.175","Text":"That\u0027ll be 7 plus 7 divided by 2 and that equals to 7."},{"Start":"03:21.175 ","End":"03:24.260","Text":"That would be the median."},{"Start":"03:24.260 ","End":"03:26.520","Text":"As we said, that\u0027s the mode,"},{"Start":"03:26.520 ","End":"03:33.140","Text":"that\u0027s the median and we\u0027re also asked about the average, that\u0027s x bar."},{"Start":"03:33.140 ","End":"03:41.135","Text":"X bar is defined as the sum of all my observations divided by the number of observations."},{"Start":"03:41.135 ","End":"03:51.440","Text":"That equals to 7 plus 6 plus 8 plus 9 plus and so on and so forth, divided by 20."},{"Start":"03:51.440 ","End":"03:55.635","Text":"Now that equals to 6.9."},{"Start":"03:55.635 ","End":"03:58.160","Text":"There we go. This is our average,"},{"Start":"03:58.160 ","End":"03:59.645","Text":"this is our median,"},{"Start":"03:59.645 ","End":"04:02.100","Text":"and this is our mode."}],"ID":13310},{"Watched":false,"Name":"Exercise 2","Duration":"4m 6s","ChapterTopicVideoID":12827,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12827.jpeg","UploadDate":"2018-07-30T17:27:01.6300000","DurationForVideoObject":"PT4M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this question, we\u0027re checking the number of rooms"},{"Start":"00:02.820 ","End":"00:05.910","Text":"in each apartment in a building with 5 apartments,"},{"Start":"00:05.910 ","End":"00:10.920","Text":"and the average rooms per apartment is given to us as 3.8."},{"Start":"00:10.920 ","End":"00:15.930","Text":"Now, the number of rooms in 4 out of the 5 apartments is as follows as 4,"},{"Start":"00:15.930 ","End":"00:17.565","Text":"3, 4, and 5."},{"Start":"00:17.565 ","End":"00:21.345","Text":"We\u0027re asked how many rooms are there in the fifth apartment?"},{"Start":"00:21.345 ","End":"00:23.085","Text":"Well, first of all,"},{"Start":"00:23.085 ","End":"00:25.575","Text":"let\u0027s define a variable."},{"Start":"00:25.575 ","End":"00:32.205","Text":"A variable would be the number of rooms per apartment."},{"Start":"00:32.205 ","End":"00:36.120","Text":"Now, how many observations do we have here?"},{"Start":"00:36.120 ","End":"00:40.835","Text":"That\u0027s our n, that equals to 5. What else are we given?"},{"Start":"00:40.835 ","End":"00:42.560","Text":"We\u0027re given that X bar,"},{"Start":"00:42.560 ","End":"00:45.380","Text":"the average number of rooms per apartment,"},{"Start":"00:45.380 ","End":"00:48.365","Text":"that equals to 3.8."},{"Start":"00:48.365 ","End":"00:51.235","Text":"How do we define an average?"},{"Start":"00:51.235 ","End":"00:53.540","Text":"X bar that equals to the sum of"},{"Start":"00:53.540 ","End":"00:58.280","Text":"the observations divided by the number of the observations."},{"Start":"00:58.280 ","End":"01:00.305","Text":"What do we have here?"},{"Start":"01:00.305 ","End":"01:02.870","Text":"What\u0027s the sum of the observations?"},{"Start":"01:02.870 ","End":"01:09.060","Text":"That\u0027s 4 plus 3 plus 4 plus 5 plus,"},{"Start":"01:09.060 ","End":"01:12.720","Text":"what\u0027s the fifth number here?"},{"Start":"01:12.720 ","End":"01:15.345","Text":"We\u0027ll call that X _5."},{"Start":"01:15.345 ","End":"01:19.500","Text":"Divided by the number of the observations."},{"Start":"01:19.500 ","End":"01:21.990","Text":"We have 5 observations here."},{"Start":"01:21.990 ","End":"01:26.105","Text":"Now, that has to equal to 3.8,"},{"Start":"01:26.105 ","End":"01:28.100","Text":"that\u0027s given to us right here."},{"Start":"01:28.100 ","End":"01:30.935","Text":"Now, if that\u0027s the case,"},{"Start":"01:30.935 ","End":"01:33.215","Text":"let\u0027s just solve for X_5."},{"Start":"01:33.215 ","End":"01:37.045","Text":"Well, multiply both sides by 5."},{"Start":"01:37.045 ","End":"01:39.900","Text":"On 1 hand we have 19,"},{"Start":"01:39.900 ","End":"01:42.540","Text":"5 times 3.8 is 19."},{"Start":"01:42.540 ","End":"01:45.020","Text":"On this side, well,"},{"Start":"01:45.020 ","End":"01:47.780","Text":"what\u0027s 4 plus 3 plus 4 plus 5?"},{"Start":"01:47.780 ","End":"01:51.900","Text":"That\u0027s 16 plus X_5."},{"Start":"01:51.900 ","End":"01:57.735","Text":"That means that X_5 for that equals to 3."},{"Start":"01:57.735 ","End":"02:01.765","Text":"The fifth department has 3 rooms."},{"Start":"02:01.765 ","End":"02:04.820","Text":"In this section, we\u0027re asked what are the mode and"},{"Start":"02:04.820 ","End":"02:09.605","Text":"the median of the number of rooms in the apartments in the building?"},{"Start":"02:09.605 ","End":"02:11.885","Text":"Let\u0027s start with the mode."},{"Start":"02:11.885 ","End":"02:15.080","Text":"That\u0027s m_o. That\u0027s the mode."},{"Start":"02:15.080 ","End":"02:16.985","Text":"Well, how do we define the mode?"},{"Start":"02:16.985 ","End":"02:22.760","Text":"That\u0027s the value that has the highest frequency."},{"Start":"02:22.760 ","End":"02:25.535","Text":"Well, let\u0027s take a look at our observations."},{"Start":"02:25.535 ","End":"02:27.875","Text":"We have here 4,"},{"Start":"02:27.875 ","End":"02:29.630","Text":"3, 4,"},{"Start":"02:29.630 ","End":"02:31.730","Text":"5, and X_5,"},{"Start":"02:31.730 ","End":"02:34.105","Text":"we calculated that as 3."},{"Start":"02:34.105 ","End":"02:36.460","Text":"Let\u0027s take a look at 3."},{"Start":"02:36.460 ","End":"02:38.120","Text":"How many 3s do we have?"},{"Start":"02:38.120 ","End":"02:41.275","Text":"I have 2 3s. And how many 4s do I have?"},{"Start":"02:41.275 ","End":"02:43.615","Text":"I have 2 4s and I have only 1 5."},{"Start":"02:43.615 ","End":"02:46.240","Text":"The mode here is not 1 number,"},{"Start":"02:46.240 ","End":"02:47.885","Text":"it\u0027s 2 numbers,"},{"Start":"02:47.885 ","End":"02:50.885","Text":"it\u0027s 3 and 4."},{"Start":"02:50.885 ","End":"02:55.070","Text":"This then is the mode of the number of rooms."},{"Start":"02:55.070 ","End":"02:57.290","Text":"Now, what about the median?"},{"Start":"02:57.290 ","End":"02:59.045","Text":"Now for the median,"},{"Start":"02:59.045 ","End":"03:02.795","Text":"what I need to do is I need to order the observations here."},{"Start":"03:02.795 ","End":"03:05.910","Text":"That\u0027ll be 3, 3,"},{"Start":"03:05.910 ","End":"03:09.645","Text":"4, 4, and 5."},{"Start":"03:09.645 ","End":"03:12.230","Text":"Now, what else do I need to do?"},{"Start":"03:12.230 ","End":"03:18.440","Text":"Well, I know that I have an odd number of observations here."},{"Start":"03:18.440 ","End":"03:22.490","Text":"That means that the median, that md."},{"Start":"03:22.490 ","End":"03:29.280","Text":"Well that equals to X_n plus 1 divided by 2."},{"Start":"03:29.280 ","End":"03:33.045","Text":"Well, what\u0027s n? N is 5."},{"Start":"03:33.045 ","End":"03:38.070","Text":"I want X_5 plus 1 divided by 2,"},{"Start":"03:38.070 ","End":"03:39.360","Text":"that\u0027s 6 divided by 2,"},{"Start":"03:39.360 ","End":"03:41.130","Text":"that\u0027s X_3."},{"Start":"03:41.130 ","End":"03:42.660","Text":"Let\u0027s just count."},{"Start":"03:42.660 ","End":"03:45.050","Text":"What\u0027s the value of X in the third place?"},{"Start":"03:45.050 ","End":"03:46.625","Text":"Well, that\u0027s the first X,"},{"Start":"03:46.625 ","End":"03:48.050","Text":"the second X,"},{"Start":"03:48.050 ","End":"03:49.640","Text":"and here\u0027s our third X."},{"Start":"03:49.640 ","End":"03:50.720","Text":"So X_ 3,"},{"Start":"03:50.720 ","End":"03:52.310","Text":"that equals to 4."},{"Start":"03:52.310 ","End":"03:55.475","Text":"4 then would be my median,"},{"Start":"03:55.475 ","End":"03:59.910","Text":"3 and 4 would be the mode,"},{"Start":"03:59.910 ","End":"04:06.420","Text":"and 3 would be the number of rooms in the fifth apartment."}],"ID":13311},{"Watched":false,"Name":"Exercise 3","Duration":"8m 16s","ChapterTopicVideoID":12828,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12828.jpeg","UploadDate":"2018-07-30T17:27:27.6900000","DurationForVideoObject":"PT8M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this question, we\u0027re given a table that"},{"Start":"00:02.340 ","End":"00:05.040","Text":"displays the probability distribution of the number of"},{"Start":"00:05.040 ","End":"00:07.320","Text":"television sets counted for each family in"},{"Start":"00:07.320 ","End":"00:10.770","Text":"a given community and we\u0027re asked to calculate the average,"},{"Start":"00:10.770 ","End":"00:14.595","Text":"median and mode of the probability distribution."},{"Start":"00:14.595 ","End":"00:18.010","Text":"The first thing that we want to do is define a variable."},{"Start":"00:18.010 ","End":"00:20.220","Text":"What\u0027s a variable?"},{"Start":"00:20.500 ","End":"00:25.890","Text":"That\u0027ll be the number of TV sets."},{"Start":"00:25.910 ","End":"00:29.160","Text":"We\u0027ll call that x."},{"Start":"00:29.160 ","End":"00:33.170","Text":"That\u0027s our x, the number of television sets 0, 1,"},{"Start":"00:33.170 ","End":"00:36.665","Text":"2, 3, and 4 and that\u0027s f of x."},{"Start":"00:36.665 ","End":"00:38.840","Text":"That\u0027s the frequency, that\u0027s the number of"},{"Start":"00:38.840 ","End":"00:42.230","Text":"families who have this number of television sets."},{"Start":"00:42.230 ","End":"00:44.690","Text":"So 22 families have 0 television sets,"},{"Start":"00:44.690 ","End":"00:48.605","Text":"28 have 1 television set and so on and so forth."},{"Start":"00:48.605 ","End":"00:51.170","Text":"This is our frequency table and we\u0027re asked,"},{"Start":"00:51.170 ","End":"00:54.415","Text":"as we said, to calculate the average, median and mode."},{"Start":"00:54.415 ","End":"00:59.470","Text":"Let\u0027s start with the mode."},{"Start":"00:59.960 ","End":"01:03.900","Text":"That\u0027s a mode. How is the mode defined?"},{"Start":"01:03.900 ","End":"01:10.805","Text":"The mode is defined as the value of x that has the highest frequency."},{"Start":"01:10.805 ","End":"01:14.405","Text":"Let\u0027s take a look at what our highest frequency is."},{"Start":"01:14.405 ","End":"01:16.400","Text":"Well, here that\u0027s 28."},{"Start":"01:16.400 ","End":"01:20.210","Text":"So 28 days isn\u0027t our mode, but 1 is,"},{"Start":"01:20.210 ","End":"01:24.255","Text":"that\u0027s the value of x that has the highest frequency."},{"Start":"01:24.255 ","End":"01:27.395","Text":"Our mode here is 1."},{"Start":"01:27.395 ","End":"01:32.110","Text":"Now, what about the average?"},{"Start":"01:32.110 ","End":"01:34.400","Text":"The average x bar."},{"Start":"01:34.400 ","End":"01:38.120","Text":"Now how is that defined when we\u0027re dealing with a frequency table?"},{"Start":"01:38.120 ","End":"01:45.875","Text":"That\u0027s defined as the sum of x times the frequency of x divided by n. Well,"},{"Start":"01:45.875 ","End":"01:48.845","Text":"first of all, what\u0027s n?"},{"Start":"01:48.845 ","End":"01:52.525","Text":"N is the sum of the frequencies."},{"Start":"01:52.525 ","End":"01:54.890","Text":"That equals to 100,"},{"Start":"01:54.890 ","End":"01:58.730","Text":"22 plus 28 and so on and so forth. That equals to 100."},{"Start":"01:58.730 ","End":"02:02.585","Text":"All we have to do now is to calculate this expression."},{"Start":"02:02.585 ","End":"02:07.980","Text":"Well, that\u0027s 0 times 22 plus 1"},{"Start":"02:07.980 ","End":"02:14.520","Text":"times 28 plus 2 times 18,"},{"Start":"02:14.520 ","End":"02:18.210","Text":"plus 3 times 22,"},{"Start":"02:18.210 ","End":"02:26.035","Text":"plus 4 times 10 and all this has to be divided by 100."},{"Start":"02:26.035 ","End":"02:30.040","Text":"Now that equals to 1.7."},{"Start":"02:30.040 ","End":"02:36.335","Text":"Here, that\u0027s the average number of TV sets per family."},{"Start":"02:36.335 ","End":"02:40.035","Text":"Now, what about the median?"},{"Start":"02:40.035 ","End":"02:43.795","Text":"The median, md, how\u0027s that defined?"},{"Start":"02:43.795 ","End":"02:47.335","Text":"Well, we have an even number of observations."},{"Start":"02:47.335 ","End":"02:53.020","Text":"So that means that the median is calculated as x_n divided by"},{"Start":"02:53.020 ","End":"02:59.585","Text":"2 plus x_n divided by 2 plus 1 divided by 2."},{"Start":"02:59.585 ","End":"03:06.365","Text":"All that means is we\u0027re taking the middle 2 observations and taking their average."},{"Start":"03:06.365 ","End":"03:09.905","Text":"Now, in order to calculate this, well,"},{"Start":"03:09.905 ","End":"03:13.895","Text":"we will be using the cumulative frequency here."},{"Start":"03:13.895 ","End":"03:16.820","Text":"Let\u0027s calculate the cumulative frequency here."},{"Start":"03:16.820 ","End":"03:22.625","Text":"Well, that will be 22 for this category and 22 plus 28, that\u0027s 50."},{"Start":"03:22.625 ","End":"03:25.445","Text":"50 plus 18, that\u0027s 68."},{"Start":"03:25.445 ","End":"03:29.285","Text":"That will be 90 and that will be 100."},{"Start":"03:29.285 ","End":"03:34.595","Text":"Now, let\u0027s take a look at what this is."},{"Start":"03:34.595 ","End":"03:37.045","Text":"X_n divided by 2."},{"Start":"03:37.045 ","End":"03:38.850","Text":"Well, n is 100,"},{"Start":"03:38.850 ","End":"03:42.100","Text":"so x_100 divided by 200,"},{"Start":"03:42.100 ","End":"03:43.580","Text":"100 divided by 2 is 50."},{"Start":"03:43.580 ","End":"03:50.360","Text":"So that\u0027ll be x_50 plus x_51 divided by 2."},{"Start":"03:50.360 ","End":"03:53.150","Text":"Now, what\u0027s x_50?"},{"Start":"03:53.150 ","End":"03:57.460","Text":"Well, if we take all the families and we put them in a row,"},{"Start":"03:57.460 ","End":"04:02.495","Text":"so the first 22 families will have no television sets."},{"Start":"04:02.495 ","End":"04:04.925","Text":"The next 28 families will have"},{"Start":"04:04.925 ","End":"04:12.560","Text":"1 television set but family number 50 is still in this category."},{"Start":"04:12.560 ","End":"04:16.655","Text":"So x_50 would be equal to 1."},{"Start":"04:16.655 ","End":"04:19.280","Text":"Now what about the 51st family?"},{"Start":"04:19.280 ","End":"04:23.210","Text":"Well, that goes to the next category right here."},{"Start":"04:23.210 ","End":"04:26.120","Text":"That means that x_51,"},{"Start":"04:26.120 ","End":"04:28.390","Text":"well that equals to 2."},{"Start":"04:28.390 ","End":"04:33.605","Text":"We\u0027re dealing here with 1 plus 2 divided by 2,"},{"Start":"04:33.605 ","End":"04:35.660","Text":"and that equals to 1.5."},{"Start":"04:35.660 ","End":"04:38.735","Text":"Our median is 1.5."},{"Start":"04:38.735 ","End":"04:41.630","Text":"Now, what\u0027s the definition of the median?"},{"Start":"04:41.630 ","End":"04:48.155","Text":"Well, that\u0027s the value where half of the families fall below this value,"},{"Start":"04:48.155 ","End":"04:51.020","Text":"and half of the families are above this value with"},{"Start":"04:51.020 ","End":"04:55.575","Text":"respect to the number of television sets and this is exactly what we have here."},{"Start":"04:55.575 ","End":"05:01.370","Text":"50 percent or 50 families are in these 2 categories,"},{"Start":"05:01.370 ","End":"05:05.780","Text":"and 50 families are in these categories right here."},{"Start":"05:05.780 ","End":"05:09.230","Text":"That means that the median has to fall"},{"Start":"05:09.230 ","End":"05:14.465","Text":"between 1 and 2 and that\u0027s exactly the average, 1.5."},{"Start":"05:14.465 ","End":"05:18.215","Text":"So 1.5 is the median."},{"Start":"05:18.215 ","End":"05:24.690","Text":"The average is 1.7 and the mode is 1."},{"Start":"05:24.760 ","End":"05:29.210","Text":"In this section, we\u0027re asked to explain without any calculation,"},{"Start":"05:29.210 ","End":"05:34.790","Text":"how each measure that we calculated in section a would change if some of the families,"},{"Start":"05:34.790 ","End":"05:39.830","Text":"not all of them, that didn\u0027t have a television set up till now were to buy 1."},{"Start":"05:39.830 ","End":"05:42.770","Text":"That means that we\u0027re taking some of the families in"},{"Start":"05:42.770 ","End":"05:46.730","Text":"this category and we\u0027re moving them to this category right here."},{"Start":"05:46.730 ","End":"05:49.970","Text":"That means that the frequency here would"},{"Start":"05:49.970 ","End":"05:54.635","Text":"decrease and the frequency here would increase. That\u0027s all."},{"Start":"05:54.635 ","End":"05:58.325","Text":"Now, let\u0027s take a look at the average."},{"Start":"05:58.325 ","End":"05:59.990","Text":"What would happen to that?"},{"Start":"05:59.990 ","End":"06:02.545","Text":"That would change,"},{"Start":"06:02.545 ","End":"06:04.205","Text":"and how would it change?"},{"Start":"06:04.205 ","End":"06:06.535","Text":"It would increase."},{"Start":"06:06.535 ","End":"06:09.785","Text":"Now, why would that be?"},{"Start":"06:09.785 ","End":"06:15.380","Text":"Well, in essence, what we\u0027re doing is we\u0027re changing the x value of the families,"},{"Start":"06:15.380 ","End":"06:18.340","Text":"the number of television sets per family."},{"Start":"06:18.340 ","End":"06:20.670","Text":"If we have values,"},{"Start":"06:20.670 ","End":"06:25.430","Text":"x values that will increase then the average also has to increase."},{"Start":"06:25.430 ","End":"06:27.800","Text":"Just take a look at the formula."},{"Start":"06:27.800 ","End":"06:31.310","Text":"X bar equals to the sum of x times"},{"Start":"06:31.310 ","End":"06:35.850","Text":"the frequency of the x divided by n. Now here, n is 100,"},{"Start":"06:35.850 ","End":"06:41.329","Text":"that remains a constant and the sum of x times f where x increases,"},{"Start":"06:41.329 ","End":"06:44.075","Text":"well, that means that the numerator increases as well."},{"Start":"06:44.075 ","End":"06:47.390","Text":"So that means that x bar,"},{"Start":"06:47.390 ","End":"06:49.700","Text":"the average also increases."},{"Start":"06:49.700 ","End":"06:53.675","Text":"Now, let\u0027s take a look at the median."},{"Start":"06:53.675 ","End":"06:57.304","Text":"Well, the median does not change."},{"Start":"06:57.304 ","End":"06:58.925","Text":"Now, why is that?"},{"Start":"06:58.925 ","End":"07:00.050","Text":"Well, what was the median?"},{"Start":"07:00.050 ","End":"07:02.315","Text":"The median was 1.5,"},{"Start":"07:02.315 ","End":"07:08.135","Text":"right here between the second and third category right here, that\u0027s 1.5."},{"Start":"07:08.135 ","End":"07:12.110","Text":"Now, all we did is we took some families"},{"Start":"07:12.110 ","End":"07:16.880","Text":"here and we moved them from the first category to the second category,"},{"Start":"07:16.880 ","End":"07:22.185","Text":"but they\u0027re still in the lower 50 percent of the families."},{"Start":"07:22.185 ","End":"07:24.740","Text":"The median does not change."},{"Start":"07:24.740 ","End":"07:27.035","Text":"Now, what about the mode?"},{"Start":"07:27.035 ","End":"07:31.415","Text":"Well, that also does not change."},{"Start":"07:31.415 ","End":"07:36.740","Text":"Now why is that? Well, what was the mode before the change?"},{"Start":"07:36.740 ","End":"07:37.820","Text":"Well, that was very here."},{"Start":"07:37.820 ","End":"07:39.155","Text":"That was 28."},{"Start":"07:39.155 ","End":"07:42.380","Text":"That was the frequency, the highest frequency."},{"Start":"07:42.380 ","End":"07:44.330","Text":"So the mode was 1."},{"Start":"07:44.330 ","End":"07:50.495","Text":"Now what we did is we took some families here and we move them to the second category."},{"Start":"07:50.495 ","End":"07:54.410","Text":"We\u0027ve increased this value right here,"},{"Start":"07:54.410 ","End":"07:59.390","Text":"the second category, we\u0027ve increased the frequency in the second category."},{"Start":"07:59.390 ","End":"08:04.890","Text":"That means that the mode is still 1."},{"Start":"08:04.890 ","End":"08:07.535","Text":"Again, the mode doesn\u0027t change,"},{"Start":"08:07.535 ","End":"08:09.380","Text":"the median doesn\u0027t change,"},{"Start":"08:09.380 ","End":"08:16.800","Text":"but the average does change because we did increase the number of TV sets."}],"ID":13312},{"Watched":false,"Name":"Exercise 4","Duration":"6m 39s","ChapterTopicVideoID":12829,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12829.jpeg","UploadDate":"2018-07-30T17:27:51.2770000","DurationForVideoObject":"PT6M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"In this question, we\u0027re given a table that displays the number of"},{"Start":"00:02.880 ","End":"00:05.970","Text":"cars owned by a family in a community and we\u0027re asked,"},{"Start":"00:05.970 ","End":"00:08.624","Text":"how many families are there in the community?"},{"Start":"00:08.624 ","End":"00:11.820","Text":"Well, first of all, what\u0027s my variable?"},{"Start":"00:11.820 ","End":"00:16.590","Text":"The variable, that\u0027s the number of"},{"Start":"00:16.590 ","End":"00:22.820","Text":"cars that are owned in a family and that\u0027ll be our x value."},{"Start":"00:22.820 ","End":"00:25.830","Text":"Here\u0027s x and here\u0027s small f of"},{"Start":"00:25.830 ","End":"00:29.720","Text":"x and we\u0027re asked how many families are there in the community?"},{"Start":"00:29.720 ","End":"00:34.295","Text":"Well, all we have to do is we have to sum up the frequency."},{"Start":"00:34.295 ","End":"00:42.620","Text":"That means that we\u0027re looking at 65 plus 150 plus 220"},{"Start":"00:42.620 ","End":"00:51.450","Text":"plus 140 plus 55 and that equals to 630."},{"Start":"00:51.450 ","End":"00:57.275","Text":"That\u0027ll be our n. The next question is,"},{"Start":"00:57.275 ","End":"01:03.410","Text":"what proportion of families in the community have at most 2 cars?"},{"Start":"01:03.410 ","End":"01:06.410","Text":"Let\u0027s just see where we\u0027re at."},{"Start":"01:06.410 ","End":"01:11.329","Text":"Here, I\u0027ve copied the table vertically."},{"Start":"01:11.329 ","End":"01:13.910","Text":"We have x from 1-5,"},{"Start":"01:13.910 ","End":"01:16.010","Text":"that\u0027s the values of x. X,"},{"Start":"01:16.010 ","End":"01:17.360","Text":"as we said, is the number of cars."},{"Start":"01:17.360 ","End":"01:20.104","Text":"Here is the frequency of families."},{"Start":"01:20.104 ","End":"01:22.730","Text":"We have 65 families who have 1 car,"},{"Start":"01:22.730 ","End":"01:24.620","Text":"150 families who have 2 cars,"},{"Start":"01:24.620 ","End":"01:25.790","Text":"and so on and so forth."},{"Start":"01:25.790 ","End":"01:31.680","Text":"Again, we\u0027re asked, what proportion of families in the community have at most 2 cars?"},{"Start":"01:31.680 ","End":"01:37.455","Text":"For that, we\u0027ll need the cumulative frequency, big F of x."},{"Start":"01:37.455 ","End":"01:39.764","Text":"That will be 65 here."},{"Start":"01:39.764 ","End":"01:45.840","Text":"That\u0027s 65 plus 150, that\u0027s 215."},{"Start":"01:45.840 ","End":"01:49.560","Text":"Here, that\u0027ll be 435,"},{"Start":"01:49.560 ","End":"01:52.155","Text":"that\u0027s 215 plus 220."},{"Start":"01:52.155 ","End":"01:57.720","Text":"435 plus 140, that\u0027s 575."},{"Start":"01:57.720 ","End":"02:02.470","Text":"575 plus 55, that\u0027s 630."},{"Start":"02:02.570 ","End":"02:10.460","Text":"Now, let\u0027s take a look at the relative cumulative frequency."},{"Start":"02:10.460 ","End":"02:14.980","Text":"That means that we\u0027re looking at big F of x divided"},{"Start":"02:14.980 ","End":"02:19.775","Text":"by n. Let\u0027s just multiply that by a 100 to get a percentage."},{"Start":"02:19.775 ","End":"02:26.180","Text":"Here we have 65 divided by 630 times"},{"Start":"02:26.180 ","End":"02:32.880","Text":"100 and that equals to 10.32 percent."},{"Start":"02:32.880 ","End":"02:35.075","Text":"Now, let\u0027s take a look at this."},{"Start":"02:35.075 ","End":"02:40.835","Text":"That\u0027s 215 divided by 630 times 100,"},{"Start":"02:40.835 ","End":"02:46.760","Text":"and that equals to 34.13 percent,"},{"Start":"02:46.760 ","End":"02:47.975","Text":"and that\u0027s our answer."},{"Start":"02:47.975 ","End":"02:54.155","Text":"We\u0027re asked about the proportional families in the community that have at most 2 cars."},{"Start":"02:54.155 ","End":"02:57.106","Text":"Well, that\u0027s this category right here."},{"Start":"02:57.106 ","End":"03:01.970","Text":"The answer is 34.13%."},{"Start":"03:01.970 ","End":"03:05.270","Text":"In the next section, we\u0027re asked to calculate the average,"},{"Start":"03:05.270 ","End":"03:07.550","Text":"the median, and the mode."},{"Start":"03:07.550 ","End":"03:10.835","Text":"Let\u0027s take a look at this table here."},{"Start":"03:10.835 ","End":"03:13.010","Text":"We\u0027ll start with some mode."},{"Start":"03:13.010 ","End":"03:15.560","Text":"Now, what\u0027s the definition of the mode?"},{"Start":"03:15.560 ","End":"03:19.940","Text":"That\u0027s the x value that has the highest frequency."},{"Start":"03:19.940 ","End":"03:22.345","Text":"Well, where\u0027s the highest frequency here?"},{"Start":"03:22.345 ","End":"03:23.955","Text":"That\u0027s small f of x."},{"Start":"03:23.955 ","End":"03:26.805","Text":"Well, that\u0027s here, that\u0027s 220."},{"Start":"03:26.805 ","End":"03:28.770","Text":"What\u0027s the x value?"},{"Start":"03:28.770 ","End":"03:33.110","Text":"That\u0027s 3. The mode here would be equal to 3."},{"Start":"03:33.110 ","End":"03:36.575","Text":"Now, what about the average?"},{"Start":"03:36.575 ","End":"03:39.635","Text":"Well, the average in a frequency table,"},{"Start":"03:39.635 ","End":"03:46.325","Text":"that equals to the sum of x times the frequency of x divided by n. Well,"},{"Start":"03:46.325 ","End":"03:53.270","Text":"that equals to 1 times 65 plus 2 times 150 plus,"},{"Start":"03:53.270 ","End":"03:58.610","Text":"and so on and so forth, divided by 630."},{"Start":"03:58.610 ","End":"04:04.880","Text":"Again, that\u0027s 1 times 65 plus 2 times 150 plus 3 times 220 plus 4 times"},{"Start":"04:04.880 ","End":"04:11.210","Text":"140 plus 5 times 55 divided by 630 divided by n. Now,"},{"Start":"04:11.210 ","End":"04:15.755","Text":"that equals to 2.95."},{"Start":"04:15.755 ","End":"04:19.040","Text":"This then would be the average."},{"Start":"04:19.040 ","End":"04:23.840","Text":"On average, a family has 2.95 cars."},{"Start":"04:23.840 ","End":"04:28.165","Text":"Now, let\u0027s take a look at the median."},{"Start":"04:28.165 ","End":"04:35.285","Text":"Well, the median is defined as the x value"},{"Start":"04:35.285 ","End":"04:38.450","Text":"where 50 percent of the families fall"},{"Start":"04:38.450 ","End":"04:42.875","Text":"below that value and 50 percent fall above that value."},{"Start":"04:42.875 ","End":"04:47.345","Text":"Now, since we have an even number of families,"},{"Start":"04:47.345 ","End":"04:51.785","Text":"then the median is calculated as X sub n divided by"},{"Start":"04:51.785 ","End":"04:58.520","Text":"2 plus X sub n divided by 2 plus 1 divided by 2."},{"Start":"04:58.520 ","End":"05:08.220","Text":"Now, again, that\u0027s the middle 2 observations or the middle 2 families divided by 2."},{"Start":"05:08.220 ","End":"05:13.000","Text":"That\u0027s the average of the number of cars of the middle 2 families."},{"Start":"05:13.000 ","End":"05:15.750","Text":"What\u0027s n divided by 2?"},{"Start":"05:15.750 ","End":"05:17.370","Text":"Well, n divided by 2,"},{"Start":"05:17.370 ","End":"05:26.400","Text":"that\u0027s 630 divided by 2 and that equals to 315."},{"Start":"05:26.400 ","End":"05:33.890","Text":"If I would place all 630 families in a row, well,"},{"Start":"05:33.890 ","End":"05:37.115","Text":"the first 65 families would have 1 car,"},{"Start":"05:37.115 ","End":"05:41.585","Text":"the next 150 families would have 2 cars and so on and so forth."},{"Start":"05:41.585 ","End":"05:46.160","Text":"Where would family number 315 be?"},{"Start":"05:46.160 ","End":"05:49.125","Text":"Well, that would be right here."},{"Start":"05:49.125 ","End":"05:51.330","Text":"That\u0027s 3."},{"Start":"05:51.330 ","End":"05:55.400","Text":"We are looking then at X sub"},{"Start":"05:55.400 ","End":"06:03.090","Text":"315 plus X sub 316 divided by 2."},{"Start":"06:03.090 ","End":"06:06.990","Text":"That means that we\u0027re looking at 3 plus,"},{"Start":"06:06.990 ","End":"06:11.395","Text":"X_315, that\u0027s right here in this category."},{"Start":"06:11.395 ","End":"06:20.445","Text":"The families in this category from place 216 on to 435,"},{"Start":"06:20.445 ","End":"06:22.485","Text":"well, they all have 3 cars."},{"Start":"06:22.485 ","End":"06:29.360","Text":"Family number 315 has 3 cars and family number 316 has 3 cars as well."},{"Start":"06:29.360 ","End":"06:33.230","Text":"That\u0027ll be 3 plus 3 divided by 2, that equals to 3."},{"Start":"06:33.230 ","End":"06:36.275","Text":"That means that the median here,"},{"Start":"06:36.275 ","End":"06:38.640","Text":"that equals to 3."}],"ID":13313},{"Watched":false,"Name":"Exercise 5","Duration":"2m 44s","ChapterTopicVideoID":12830,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12830.jpeg","UploadDate":"2018-07-30T17:28:01.6670000","DurationForVideoObject":"PT2M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this question, we have a teacher that teaches 2 classes."},{"Start":"00:03.180 ","End":"00:05.910","Text":"Now he plots the probability distribution of"},{"Start":"00:05.910 ","End":"00:10.245","Text":"the marks for both classes on the same set of coordinates."},{"Start":"00:10.245 ","End":"00:12.495","Text":"We have here,"},{"Start":"00:12.495 ","End":"00:16.275","Text":"the probability distributions of Class number 1,"},{"Start":"00:16.275 ","End":"00:18.810","Text":"that\u0027s Class number 1 and Class number 2."},{"Start":"00:18.810 ","End":"00:21.189","Text":"Here, that\u0027s our x-axis,"},{"Start":"00:21.189 ","End":"00:23.010","Text":"that\u0027s the marks right here."},{"Start":"00:23.010 ","End":"00:25.995","Text":"We\u0027re asked to select the correct answer."},{"Start":"00:25.995 ","End":"00:30.075","Text":"The mode of Class 1 is higher than the mode of Class 2."},{"Start":"00:30.075 ","End":"00:33.375","Text":"The mode of Class 2 is higher than the mode of Class 1."},{"Start":"00:33.375 ","End":"00:35.535","Text":"Both classes have the same mode,"},{"Start":"00:35.535 ","End":"00:40.820","Text":"and there\u0027s not enough information to determine which class has the higher mode."},{"Start":"00:40.820 ","End":"00:43.580","Text":"Now, if you\u0027re like a lot of people,"},{"Start":"00:43.580 ","End":"00:47.240","Text":"then you choose the first answer right here,"},{"Start":"00:47.240 ","End":"00:50.240","Text":"that the mode of Class 1 is higher than the mode of Class 2."},{"Start":"00:50.240 ","End":"00:52.910","Text":"But that would be the wrong answer, and why is that?"},{"Start":"00:52.910 ","End":"00:57.890","Text":"Well, you\u0027d be confused between the mode and the frequency."},{"Start":"00:57.890 ","End":"01:02.030","Text":"Now, let\u0027s take a look at Class 1, and where\u0027s the mode?"},{"Start":"01:02.030 ","End":"01:04.040","Text":"What\u0027s the definition of the mode?"},{"Start":"01:04.040 ","End":"01:09.485","Text":"The mode is the x value that has the highest frequency."},{"Start":"01:09.485 ","End":"01:11.750","Text":"Now, in Class number 1,"},{"Start":"01:11.750 ","End":"01:13.775","Text":"here\u0027s the highest frequency."},{"Start":"01:13.775 ","End":"01:21.680","Text":"But this then x_1 is the mode of class number 1. Why is that?"},{"Start":"01:21.680 ","End":"01:25.849","Text":"Because the value x_1 has the highest frequency."},{"Start":"01:25.849 ","End":"01:29.000","Text":"The same argument for Class number 2 here."},{"Start":"01:29.000 ","End":"01:33.481","Text":"In Class number 2, here\u0027s the highest frequency right here."},{"Start":"01:33.481 ","End":"01:37.505","Text":"But here, that\u0027s the mode of Class 2."},{"Start":"01:37.505 ","End":"01:41.679","Text":"That\u0027s the x value that has the highest frequency."},{"Start":"01:41.679 ","End":"01:47.565","Text":"That means that we can see that x_2 is greater than x_1."},{"Start":"01:47.565 ","End":"01:49.790","Text":"The correct answer is right here."},{"Start":"01:49.790 ","End":"01:54.320","Text":"The mode of Class 2 is higher than the mode of Class 1."},{"Start":"01:54.320 ","End":"01:56.030","Text":"Now, if that\u0027s not clear,"},{"Start":"01:56.030 ","End":"01:59.900","Text":"let\u0027s just put some numeric figures on the x-axis right here."},{"Start":"01:59.900 ","End":"02:02.915","Text":"Let\u0027s say this is 50,"},{"Start":"02:02.915 ","End":"02:06.065","Text":"this is 60, 70,"},{"Start":"02:06.065 ","End":"02:09.995","Text":"80, 90, and so on and so forth."},{"Start":"02:09.995 ","End":"02:14.345","Text":"What do we see here that the mode of x_1,"},{"Start":"02:14.345 ","End":"02:17.140","Text":"that equals to 60,"},{"Start":"02:17.140 ","End":"02:19.800","Text":"and the mode of x_2,"},{"Start":"02:19.800 ","End":"02:22.005","Text":"that equals to 70."},{"Start":"02:22.005 ","End":"02:26.070","Text":"We see here that 70 is greater than 60,"},{"Start":"02:26.070 ","End":"02:28.289","Text":"60 is less than 70,"},{"Start":"02:28.289 ","End":"02:33.390","Text":"and 60 is the mode for Class number 1,"},{"Start":"02:33.390 ","End":"02:36.165","Text":"70 is the mode for class number 2."},{"Start":"02:36.165 ","End":"02:39.330","Text":"So b, that\u0027s the right answer,"},{"Start":"02:39.330 ","End":"02:44.640","Text":"the mode of Class 2 is higher than the mode of Class 1."}],"ID":13314},{"Watched":false,"Name":"Exercise 6","Duration":"10m 1s","ChapterTopicVideoID":12831,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12831.jpeg","UploadDate":"2018-07-30T17:28:31.1330000","DurationForVideoObject":"PT10M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this question, we\u0027re given a table that"},{"Start":"00:02.070 ","End":"00:04.875","Text":"displays the number of television sets per family."},{"Start":"00:04.875 ","End":"00:07.380","Text":"If there are 200 families in the community and"},{"Start":"00:07.380 ","End":"00:11.520","Text":"the average number of TV sets per family is 1.5."},{"Start":"00:11.520 ","End":"00:14.010","Text":"We\u0027re asked to complete the table."},{"Start":"00:14.010 ","End":"00:16.770","Text":"That means that we see here 2 columns,"},{"Start":"00:16.770 ","End":"00:21.840","Text":"1 column is the number of TV sets and the second column is the number of families,"},{"Start":"00:21.840 ","End":"00:26.700","Text":"and we have 2 cells here that are missing and we have to calculate their values."},{"Start":"00:26.700 ","End":"00:29.010","Text":"Now, in descriptive statistics,"},{"Start":"00:29.010 ","End":"00:31.650","Text":"we like to define what we\u0027re dealing with."},{"Start":"00:31.650 ","End":"00:35.830","Text":"The first thing that we want to define is the group."},{"Start":"00:35.830 ","End":"00:37.940","Text":"What\u0027s the group that we\u0027re dealing with?"},{"Start":"00:37.940 ","End":"00:42.705","Text":"Well, that\u0027s the families in the community."},{"Start":"00:42.705 ","End":"00:46.370","Text":"What\u0027s the variable that we\u0027re looking at?"},{"Start":"00:46.370 ","End":"00:48.290","Text":"Well, what changes from family to family?"},{"Start":"00:48.290 ","End":"00:51.030","Text":"Well, that\u0027s the number of TV sets."},{"Start":"00:52.690 ","End":"00:57.409","Text":"That\u0027ll be our x, that\u0027s the number of TV sets,"},{"Start":"00:57.409 ","End":"01:01.385","Text":"and the number of families that\u0027s the frequency, that\u0027s f of x."},{"Start":"01:01.385 ","End":"01:04.900","Text":"We\u0027re also given that we have 200 families,"},{"Start":"01:04.900 ","End":"01:07.009","Text":"so the total number of families here,"},{"Start":"01:07.009 ","End":"01:09.560","Text":"that equals to 200."},{"Start":"01:09.560 ","End":"01:14.300","Text":"Now, let\u0027s define this value right here as a."},{"Start":"01:14.300 ","End":"01:16.310","Text":"What\u0027s this value?"},{"Start":"01:16.310 ","End":"01:22.310","Text":"Well, that\u0027s 110 minus a. Now why is that?"},{"Start":"01:22.310 ","End":"01:25.970","Text":"Well, this is 28 plus 62,"},{"Start":"01:25.970 ","End":"01:27.680","Text":"that equals to 90."},{"Start":"01:27.680 ","End":"01:34.525","Text":"So 200 minus 90 minus a,"},{"Start":"01:34.525 ","End":"01:39.795","Text":"that equals to 110 minus a and that\u0027s this value right here."},{"Start":"01:39.795 ","End":"01:43.120","Text":"We\u0027re looking to calculate these 2 values right here."},{"Start":"01:43.120 ","End":"01:46.790","Text":"We haven\u0027t used this piece of information."},{"Start":"01:46.790 ","End":"01:51.185","Text":"This means that x bar the average,"},{"Start":"01:51.185 ","End":"01:56.720","Text":"that\u0027s the average number of television sets, that equals to 1.5."},{"Start":"01:56.720 ","End":"02:03.245","Text":"How do we calculate the average of a frequency table of a discrete frequency table?"},{"Start":"02:03.245 ","End":"02:12.050","Text":"That equals to the sum of x times the frequency divided by n. What is that equal to?"},{"Start":"02:12.050 ","End":"02:15.429","Text":"The sum of x times the frequency,"},{"Start":"02:15.429 ","End":"02:20.480","Text":"that\u0027s 0 times 28 plus 1"},{"Start":"02:20.480 ","End":"02:26.045","Text":"times 62 plus 2 times"},{"Start":"02:26.045 ","End":"02:32.920","Text":"a plus 3 times 110 minus a."},{"Start":"02:32.920 ","End":"02:37.730","Text":"All that has to be divided by 200."},{"Start":"02:37.730 ","End":"02:42.455","Text":"That equals to what?"},{"Start":"02:42.455 ","End":"02:48.650","Text":"That equals to 62 plus 2a"},{"Start":"02:48.650 ","End":"02:55.740","Text":"plus 330 minus 3a,"},{"Start":"02:55.740 ","End":"03:01.210","Text":"and that has to be divided by 200."},{"Start":"03:02.990 ","End":"03:10.805","Text":"Let\u0027s continue. This then has to equal to 1.5 so let\u0027s multiply both sides by 200."},{"Start":"03:10.805 ","End":"03:13.740","Text":"We get 300 here,"},{"Start":"03:13.740 ","End":"03:22.425","Text":"that equals to 62 plus 2a plus 330 minus 3a."},{"Start":"03:22.425 ","End":"03:28.820","Text":"That equals to the following."},{"Start":"03:28.820 ","End":"03:36.730","Text":"300 that has to equal to 392 minus a."},{"Start":"03:36.730 ","End":"03:42.280","Text":"That means that a equals to 92."},{"Start":"03:42.280 ","End":"03:49.600","Text":"Now we know that here a equals to 92 and 110 minus a or minus 92,"},{"Start":"03:49.600 ","End":"03:52.525","Text":"that equals to 18. There we go."},{"Start":"03:52.525 ","End":"03:54.890","Text":"We completed the table."},{"Start":"03:55.040 ","End":"03:57.820","Text":"In this section we\u0027re asked to find the mode,"},{"Start":"03:57.820 ","End":"04:00.440","Text":"the mid-range, and the median."},{"Start":"04:01.350 ","End":"04:07.465","Text":"I copied here the completed frequency table."},{"Start":"04:07.465 ","End":"04:09.550","Text":"Let\u0027s take a look at the mode."},{"Start":"04:09.550 ","End":"04:11.365","Text":"What\u0027s the definition of the mode?"},{"Start":"04:11.365 ","End":"04:16.090","Text":"The mode is the x value that has the highest frequency."},{"Start":"04:16.090 ","End":"04:21.175","Text":"We have to go to the frequency column right here and look for the highest value,"},{"Start":"04:21.175 ","End":"04:24.055","Text":"that\u0027s this value right here, 92."},{"Start":"04:24.055 ","End":"04:29.290","Text":"Then we have to see what\u0027s the x value that has the size frequency?"},{"Start":"04:29.290 ","End":"04:30.985","Text":"Well, that\u0027s 2."},{"Start":"04:30.985 ","End":"04:33.940","Text":"The mode here would be 2."},{"Start":"04:33.940 ","End":"04:37.820","Text":"What about the mid-range?"},{"Start":"04:37.820 ","End":"04:41.214","Text":"What\u0027s the definition of the mid-range?"},{"Start":"04:41.214 ","End":"04:43.615","Text":"That\u0027s the X_max,"},{"Start":"04:43.615 ","End":"04:47.325","Text":"the maximum value of X minus X_min,"},{"Start":"04:47.325 ","End":"04:51.180","Text":"that\u0027s a minimum value of x divided by 2."},{"Start":"04:51.180 ","End":"05:00.295","Text":"We take the 2 edges of the distribution and we take their average."},{"Start":"05:00.295 ","End":"05:02.880","Text":"The maximum value of x,"},{"Start":"05:02.880 ","End":"05:07.820","Text":"that\u0027s 3 minus the minimum value that\u0027s 0 divided by 2."},{"Start":"05:07.820 ","End":"05:11.810","Text":"Now that equals to 1.5."},{"Start":"05:11.810 ","End":"05:14.480","Text":"What about the median?"},{"Start":"05:14.480 ","End":"05:17.090","Text":"Well, since we have an even number of"},{"Start":"05:17.090 ","End":"05:20.585","Text":"observations and the definition of the median would be"},{"Start":"05:20.585 ","End":"05:28.310","Text":"X_n divided by 2 plus X_n divided by 2 plus 1 divided by 2."},{"Start":"05:28.310 ","End":"05:30.905","Text":"All that means again is that we\u0027re taking"},{"Start":"05:30.905 ","End":"05:34.280","Text":"the 2 middle observations and taking their average."},{"Start":"05:34.280 ","End":"05:37.880","Text":"In order to calculate the x value,"},{"Start":"05:37.880 ","End":"05:43.805","Text":"then we have to go to the cumulative frequency of this distribution."},{"Start":"05:43.805 ","End":"05:48.230","Text":"Here that\u0027ll be 28. Now 28 plus 62, that\u0027s 90,"},{"Start":"05:48.230 ","End":"05:54.930","Text":"90 plus 92 that\u0027s 182 and 182 plus 18, that\u0027s 200."},{"Start":"05:55.550 ","End":"05:59.300","Text":"What\u0027s X_ n divided by 2?"},{"Start":"05:59.300 ","End":"06:01.325","Text":"Well, n divided by 2,"},{"Start":"06:01.325 ","End":"06:05.375","Text":"that equals to 200 divided by 2 that equals to 100."},{"Start":"06:05.375 ","End":"06:15.595","Text":"We\u0027re looking for X_100 plus X_101 divided by 2."},{"Start":"06:15.595 ","End":"06:19.460","Text":"We\u0027re looking for families or x values."},{"Start":"06:19.460 ","End":"06:22.160","Text":"Families with the number of television sets,"},{"Start":"06:22.160 ","End":"06:26.600","Text":"families that are in the 100\u0027s place and the 101\u0027s place."},{"Start":"06:26.600 ","End":"06:28.850","Text":"Now, how do we calculate that?"},{"Start":"06:28.850 ","End":"06:37.760","Text":"Let\u0027s go to the cumulative frequency and see what category is the 100 and 101\u0027s place?"},{"Start":"06:37.760 ","End":"06:42.490","Text":"It start in this category because this category goes up to 28."},{"Start":"06:42.490 ","End":"06:45.230","Text":"The second category goes up to 90."},{"Start":"06:45.230 ","End":"06:48.350","Text":"The third category goes up to 182."},{"Start":"06:48.350 ","End":"06:53.835","Text":"That means that family in place number 100,"},{"Start":"06:53.835 ","End":"06:56.645","Text":"that\u0027s in the third category right here."},{"Start":"06:56.645 ","End":"07:01.655","Text":"Now, what\u0027s the x value in this category well, that\u0027s 2."},{"Start":"07:01.655 ","End":"07:08.305","Text":"Now, what\u0027s the x value of the family that\u0027s in place number 101?"},{"Start":"07:08.305 ","End":"07:11.780","Text":"Well again, that\u0027s also in the third category."},{"Start":"07:11.780 ","End":"07:14.240","Text":"The x value here is also 2,"},{"Start":"07:14.240 ","End":"07:19.595","Text":"so it\u0027ll be 2 plus 2 divided by 2 and that equals to 2. There we go."},{"Start":"07:19.595 ","End":"07:21.115","Text":"That\u0027s the median,"},{"Start":"07:21.115 ","End":"07:23.160","Text":"this is the mid range,"},{"Start":"07:23.160 ","End":"07:25.735","Text":"and that\u0027s the mode."},{"Start":"07:25.735 ","End":"07:30.710","Text":"In this section, we\u0027re given that some of the families that had exactly 1 television set"},{"Start":"07:30.710 ","End":"07:35.000","Text":"took it out of their homes and were asked how would each measure change?"},{"Start":"07:35.000 ","End":"07:36.589","Text":"Will it increase, decrease,"},{"Start":"07:36.589 ","End":"07:38.135","Text":"or will it be unchanged?"},{"Start":"07:38.135 ","End":"07:42.455","Text":"Now we have to explain our answer without doing any calculations."},{"Start":"07:42.455 ","End":"07:44.315","Text":"Let\u0027s see where we\u0027re going with this."},{"Start":"07:44.315 ","End":"07:46.640","Text":"First of all, we have families in"},{"Start":"07:46.640 ","End":"07:51.710","Text":"this category and they took their television out of their homes."},{"Start":"07:51.710 ","End":"07:54.500","Text":"That means that now they have no television set."},{"Start":"07:54.500 ","End":"07:56.810","Text":"That means that some of the families here in"},{"Start":"07:56.810 ","End":"08:00.380","Text":"this category moved into this category right here."},{"Start":"08:00.380 ","End":"08:06.200","Text":"That means that the frequency here decreased and the frequency here increased."},{"Start":"08:06.200 ","End":"08:10.430","Text":"Now, how would that affect the mode?"},{"Start":"08:10.430 ","End":"08:13.235","Text":"If we\u0027re dealing with a mode,"},{"Start":"08:13.235 ","End":"08:15.710","Text":"well, what\u0027s the definition of a mode?"},{"Start":"08:15.710 ","End":"08:23.610","Text":"The mode is the x value where frequency is the highest."},{"Start":"08:23.930 ","End":"08:26.540","Text":"What\u0027s the highest frequency here?"},{"Start":"08:26.540 ","End":"08:31.925","Text":"That\u0027s 92. That means that the mode here would be 2."},{"Start":"08:31.925 ","End":"08:34.715","Text":"Really doesn\u0027t matter what happens here,"},{"Start":"08:34.715 ","End":"08:40.045","Text":"because the highest frequency remains unchanged."},{"Start":"08:40.045 ","End":"08:44.895","Text":"The mode here that\u0027s unchanged."},{"Start":"08:44.895 ","End":"08:47.425","Text":"Now, what about the mid range?"},{"Start":"08:47.425 ","End":"08:50.840","Text":"Well, the definition of the mid range is"},{"Start":"08:50.840 ","End":"08:57.560","Text":"the maximum value of x minus the minimum value of x divided by 2."},{"Start":"08:57.560 ","End":"09:04.100","Text":"Now here, the x values didn\u0027t change it\u0027s just the distribution of families."},{"Start":"09:04.100 ","End":"09:06.170","Text":"It\u0027s just the frequencies."},{"Start":"09:06.170 ","End":"09:12.280","Text":"We still have the maximum value of x being 3 and the minimum value of x"},{"Start":"09:12.280 ","End":"09:18.500","Text":"being 0 and so therefore the mid-range remains unchanged."},{"Start":"09:18.500 ","End":"09:22.280","Text":"That also is unchanged."},{"Start":"09:22.280 ","End":"09:25.534","Text":"Now, what happens to the median?"},{"Start":"09:25.534 ","End":"09:27.935","Text":"Well, what\u0027s the median?"},{"Start":"09:27.935 ","End":"09:32.750","Text":"The median is the x value where half of"},{"Start":"09:32.750 ","End":"09:38.135","Text":"the families are below this value and half of the families are above this value."},{"Start":"09:38.135 ","End":"09:41.520","Text":"Now, the median here is 2."},{"Start":"09:41.630 ","End":"09:46.785","Text":"Again, the redistribution of families."},{"Start":"09:46.785 ","End":"09:52.640","Text":"The movement of families from this category to this category right here has"},{"Start":"09:52.640 ","End":"10:00.040","Text":"no effect whatsoever on the median so that also is unchanged."}],"ID":13315},{"Watched":false,"Name":"Exercise 7","Duration":"2m 41s","ChapterTopicVideoID":12832,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12832.jpeg","UploadDate":"2018-07-30T17:28:38.6470000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this question, we have a table that displays"},{"Start":"00:02.730 ","End":"00:05.670","Text":"the weight in kilograms for a certain group of people."},{"Start":"00:05.670 ","End":"00:11.123","Text":"We\u0027re asked what\u0027s the average and median of the probability distribution,"},{"Start":"00:11.123 ","End":"00:13.845","Text":"so let\u0027s just define what we\u0027re looking at."},{"Start":"00:13.845 ","End":"00:16.350","Text":"What\u0027s the group that we\u0027re looking at?"},{"Start":"00:16.350 ","End":"00:19.034","Text":"That\u0027s a group of people?"},{"Start":"00:19.034 ","End":"00:21.540","Text":"What\u0027s the variable?"},{"Start":"00:21.540 ","End":"00:24.750","Text":"What changes from a person to a person?"},{"Start":"00:24.750 ","End":"00:27.580","Text":"That\u0027s the weight."},{"Start":"00:27.800 ","End":"00:30.270","Text":"We\u0027ll denote that as X,"},{"Start":"00:30.270 ","End":"00:32.670","Text":"so X would be our weight,"},{"Start":"00:32.670 ","End":"00:35.250","Text":"and f of x would be the number of people,"},{"Start":"00:35.250 ","End":"00:38.144","Text":"so we have 10 people, that\u0027s the frequency."},{"Start":"00:38.144 ","End":"00:42.440","Text":"10 people who weigh between 40 and 45 kilograms,"},{"Start":"00:42.440 ","End":"00:46.825","Text":"20 people who weigh between 45 and 50 kilograms and so on and so forth."},{"Start":"00:46.825 ","End":"00:50.900","Text":"Now, we\u0027re asked about the average and median of this distribution."},{"Start":"00:50.900 ","End":"00:52.850","Text":"Now, before we start calculating,"},{"Start":"00:52.850 ","End":"00:56.945","Text":"let\u0027s just take a look at this distribution."},{"Start":"00:56.945 ","End":"01:04.865","Text":"Well, we\u0027ll see that it\u0027s symmetrical distribution. Now why is that?"},{"Start":"01:04.865 ","End":"01:10.955","Text":"Well, let\u0027s take a look at this category right here, that\u0027s 10."},{"Start":"01:10.955 ","End":"01:15.800","Text":"The frequency of this category is 10, the lowermost category."},{"Start":"01:15.800 ","End":"01:18.005","Text":"What about the uppermost category?"},{"Start":"01:18.005 ","End":"01:19.700","Text":"But that\u0027s also 10."},{"Start":"01:19.700 ","End":"01:23.300","Text":"Now let\u0027s take a look at the width of this category right here."},{"Start":"01:23.300 ","End":"01:24.980","Text":"Well that\u0027s 45 minus 40,"},{"Start":"01:24.980 ","End":"01:29.555","Text":"that\u0027s 5, and that\u0027s 70 minus 65, that\u0027s also 5."},{"Start":"01:29.555 ","End":"01:39.305","Text":"The width of the categories right here are equal and also the frequencies."},{"Start":"01:39.305 ","End":"01:42.920","Text":"Let\u0027s take a look at the inner categories right here."},{"Start":"01:42.920 ","End":"01:45.305","Text":"Well, the next category up is 20."},{"Start":"01:45.305 ","End":"01:48.110","Text":"The frequency here of this category is"},{"Start":"01:48.110 ","End":"01:52.840","Text":"20 and the frequency of this category right here is 20 as well."},{"Start":"01:52.840 ","End":"01:56.660","Text":"Here the width of this category is 50 minus 45,"},{"Start":"01:56.660 ","End":"01:58.100","Text":"that\u0027s 5 and here,"},{"Start":"01:58.100 ","End":"02:01.885","Text":"that\u0027s 65 minus 60, that\u0027s also 5."},{"Start":"02:01.885 ","End":"02:06.405","Text":"Here we have a symmetrical distribution."},{"Start":"02:06.405 ","End":"02:12.080","Text":"Now, if the frequency is the same and the category width are the same,"},{"Start":"02:12.080 ","End":"02:15.215","Text":"then we know that the average, well,"},{"Start":"02:15.215 ","End":"02:20.090","Text":"that equals to the median.. What\u0027s that?"},{"Start":"02:20.090 ","End":"02:23.695","Text":"Well, we have the middle category right here."},{"Start":"02:23.695 ","End":"02:26.585","Text":"That\u0027ll be the middle of this category."},{"Start":"02:26.585 ","End":"02:30.212","Text":"That\u0027ll 60 minus 50 divided by 2,"},{"Start":"02:30.212 ","End":"02:33.185","Text":"that equals to 55."},{"Start":"02:33.185 ","End":"02:41.400","Text":"This then is the average and the median of this probability distribution."}],"ID":13316},{"Watched":false,"Name":"Exercise 8","Duration":"7m 16s","ChapterTopicVideoID":12833,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12833.jpeg","UploadDate":"2018-07-30T17:29:07.0830000","DurationForVideoObject":"PT7M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this question, we\u0027re given a table that shows"},{"Start":"00:02.370 ","End":"00:07.125","Text":"the probability distribution of the height in centimeters in a certain group of people."},{"Start":"00:07.125 ","End":"00:09.345","Text":"Now, in descriptive statistics,"},{"Start":"00:09.345 ","End":"00:11.580","Text":"we\u0027re interested in analyzing groups,"},{"Start":"00:11.580 ","End":"00:14.079","Text":"and here we have a specific group of people,"},{"Start":"00:14.079 ","End":"00:15.405","Text":"so that\u0027s our group,"},{"Start":"00:15.405 ","End":"00:19.380","Text":"and we\u0027re interested in analyzing that group through variables."},{"Start":"00:19.380 ","End":"00:23.760","Text":"Now, what varies from person to person here in this group? Well, that\u0027s the height."},{"Start":"00:23.760 ","End":"00:28.740","Text":"X then would be the height, and here,"},{"Start":"00:28.740 ","End":"00:34.460","Text":"that\u0027ll be the frequency of the people that have a specific height."},{"Start":"00:34.460 ","End":"00:36.680","Text":"Now, this table is given to us,"},{"Start":"00:36.680 ","End":"00:38.300","Text":"that\u0027s a frequency table,"},{"Start":"00:38.300 ","End":"00:41.990","Text":"but x is given to us in categories."},{"Start":"00:41.990 ","End":"00:44.600","Text":"We\u0027re asked now to calculate the average,"},{"Start":"00:44.600 ","End":"00:48.185","Text":"the median, and the mode of the height in this group."},{"Start":"00:48.185 ","End":"00:52.550","Text":"Let\u0027s start with the average, that\u0027s x bar."},{"Start":"00:52.550 ","End":"01:00.272","Text":"Now, that\u0027s defined as the sum of x times f of x divided by n. Well,"},{"Start":"01:00.272 ","End":"01:02.630","Text":"here, n equals to 240,"},{"Start":"01:02.630 ","End":"01:04.925","Text":"that\u0027s just the sum of all the people."},{"Start":"01:04.925 ","End":"01:07.280","Text":"Now, because we have categories,"},{"Start":"01:07.280 ","End":"01:12.420","Text":"we can\u0027t multiply x times f of x."},{"Start":"01:12.420 ","End":"01:21.105","Text":"What we need to do is we need to identify the midpoint for each category."},{"Start":"01:21.105 ","End":"01:23.550","Text":"Now, here that\u0027ll be 155,"},{"Start":"01:23.550 ","End":"01:25.785","Text":"here that\u0027s 165,"},{"Start":"01:25.785 ","End":"01:29.325","Text":"here that\u0027ll be 172.5,"},{"Start":"01:29.325 ","End":"01:34.170","Text":"that\u0027s 177.5, and that\u0027s 185."},{"Start":"01:34.170 ","End":"01:40.055","Text":"Now, we can calculate this expression right here."},{"Start":"01:40.055 ","End":"01:44.645","Text":"For x, we\u0027ll use the midpoint of the categories,"},{"Start":"01:44.645 ","End":"01:55.695","Text":"so it\u0027ll be 155 times 30 plus 165 times 40 plus 172.5"},{"Start":"01:55.695 ","End":"02:01.560","Text":"times 60 plus 177.5 times"},{"Start":"02:01.560 ","End":"02:08.055","Text":"70 plus 185 times 40,"},{"Start":"02:08.055 ","End":"02:12.815","Text":"and we have to divide that by 240."},{"Start":"02:12.815 ","End":"02:19.070","Text":"Now, that comes out to 172.6 centimeters."},{"Start":"02:19.070 ","End":"02:22.355","Text":"That\u0027s the average height of the people."},{"Start":"02:22.355 ","End":"02:26.150","Text":"Let\u0027s take a look now at the median."},{"Start":"02:26.150 ","End":"02:35.390","Text":"Now, how do we calculate the median when we have a frequency table with categories?"},{"Start":"02:35.390 ","End":"02:37.430","Text":"Well, the first thing that we need to do is we need to"},{"Start":"02:37.430 ","End":"02:44.775","Text":"identify in which category does the median lie."},{"Start":"02:44.775 ","End":"02:46.475","Text":"Well, in order to do that,"},{"Start":"02:46.475 ","End":"02:49.385","Text":"then we need the cumulative frequency,"},{"Start":"02:49.385 ","End":"02:51.905","Text":"that\u0027s F of x."},{"Start":"02:51.905 ","End":"02:53.705","Text":"Here, that\u0027ll be 30,"},{"Start":"02:53.705 ","End":"02:55.370","Text":"here 30 plus 40,"},{"Start":"02:55.370 ","End":"02:57.590","Text":"that\u0027s 70, 70 plus 60,"},{"Start":"02:57.590 ","End":"03:01.505","Text":"that\u0027s 130, 130 plus 70 that\u0027s 200,"},{"Start":"03:01.505 ","End":"03:05.165","Text":"and 200 plus 40, that\u0027s 240."},{"Start":"03:05.165 ","End":"03:07.580","Text":"Now, what\u0027s the definition of the median?"},{"Start":"03:07.580 ","End":"03:12.590","Text":"The median is the x value where half of"},{"Start":"03:12.590 ","End":"03:18.685","Text":"the people lie below this value and half of the people lie above that value."},{"Start":"03:18.685 ","End":"03:23.420","Text":"What\u0027s half of 240 because we have 240 people?"},{"Start":"03:23.420 ","End":"03:26.315","Text":"Well, 240 divided by 2,"},{"Start":"03:26.315 ","End":"03:28.700","Text":"that equals to 120."},{"Start":"03:28.700 ","End":"03:36.965","Text":"We\u0027re looking at a person that\u0027s in the place 120 out of 240."},{"Start":"03:36.965 ","End":"03:41.300","Text":"Now, that\u0027s why we calculated the cumulative frequency here."},{"Start":"03:41.300 ","End":"03:46.765","Text":"We see that place number 120 is in this category right here,"},{"Start":"03:46.765 ","End":"03:52.225","Text":"so this is the category where the median lies."},{"Start":"03:52.225 ","End":"03:55.550","Text":"Now, if that\u0027s the case,"},{"Start":"03:55.550 ","End":"03:58.265","Text":"now let\u0027s calculate the median."},{"Start":"03:58.265 ","End":"04:01.760","Text":"The median, Md, that equals to the following,"},{"Start":"04:01.760 ","End":"04:11.555","Text":"that\u0027s L_0 plus n divided by 2 minus the cumulative frequency of"},{"Start":"04:11.555 ","End":"04:16.660","Text":"x_m minus 1 divided by"},{"Start":"04:16.660 ","End":"04:23.240","Text":"f of x at m times L_1 minus L_0."},{"Start":"04:23.240 ","End":"04:24.800","Text":"Now, what does that mean?"},{"Start":"04:24.800 ","End":"04:29.780","Text":"Well, L_0, that\u0027s the lower bound of the category where the median is,"},{"Start":"04:29.780 ","End":"04:34.040","Text":"that\u0027s 170 plus, well,"},{"Start":"04:34.040 ","End":"04:37.850","Text":"n divided by 2, we calculated that, that\u0027s 120."},{"Start":"04:37.850 ","End":"04:41.780","Text":"Now, F of x at m minus 1,"},{"Start":"04:41.780 ","End":"04:46.070","Text":"that means that we\u0027re looking at the cumulative frequency"},{"Start":"04:46.070 ","End":"04:51.560","Text":"of the category preceding the category where the median is,"},{"Start":"04:51.560 ","End":"04:54.200","Text":"so if this is the category where the median is,"},{"Start":"04:54.200 ","End":"04:56.710","Text":"f of the previous category,"},{"Start":"04:56.710 ","End":"05:00.230","Text":"that 70, that\u0027s minus 70, it\u0027s right here,"},{"Start":"05:00.230 ","End":"05:07.260","Text":"divided by f of x in that category,"},{"Start":"05:07.260 ","End":"05:09.096","Text":"in the category where the median is, well,"},{"Start":"05:09.096 ","End":"05:13.625","Text":"that\u0027s 60 times,"},{"Start":"05:13.625 ","End":"05:16.130","Text":"now, L_1 minus L_0, well,"},{"Start":"05:16.130 ","End":"05:20.600","Text":"that\u0027s the upper bound and the lower bound of the category where the median is,"},{"Start":"05:20.600 ","End":"05:25.000","Text":"so that\u0027s 175 minus 170."},{"Start":"05:25.000 ","End":"05:27.290","Text":"Now, when we calculate that,"},{"Start":"05:27.290 ","End":"05:32.430","Text":"that comes out to 174.17."},{"Start":"05:33.560 ","End":"05:38.140","Text":"Now, let\u0027s take a look at the mode."},{"Start":"05:38.140 ","End":"05:42.380","Text":"Well, in order to calculate the mode, well,"},{"Start":"05:42.380 ","End":"05:50.015","Text":"this is the midpoint of the category where we have the highest density."},{"Start":"05:50.015 ","End":"05:54.590","Text":"In order to do that, we have to calculate d or density."},{"Start":"05:54.590 ","End":"05:57.980","Text":"Now, how\u0027s that defied?"},{"Start":"05:57.980 ","End":"06:07.100","Text":"The density is defined as f of x divided by L_1 minus L_0."},{"Start":"06:07.100 ","End":"06:16.230","Text":"That\u0027s means that in this category we have 30 divided by 160 minus 150,"},{"Start":"06:16.230 ","End":"06:19.760","Text":"that\u0027s 10, so that equals to 3."},{"Start":"06:19.760 ","End":"06:26.720","Text":"Here, we have 40 divided by 170 minus 160,"},{"Start":"06:26.720 ","End":"06:29.315","Text":"that\u0027s 10, that equals to 4,"},{"Start":"06:29.315 ","End":"06:31.415","Text":"and so on and so forth."},{"Start":"06:31.415 ","End":"06:34.370","Text":"Here we have 12,"},{"Start":"06:34.370 ","End":"06:38.630","Text":"we have 14, and we have 4."},{"Start":"06:38.630 ","End":"06:44.480","Text":"We can see that the category with the highest density, that\u0027s right here,"},{"Start":"06:44.480 ","End":"06:52.580","Text":"14, so that means that this then is the category with the highest density."},{"Start":"06:52.580 ","End":"06:56.360","Text":"What the midpoint right here of this category?"},{"Start":"06:56.360 ","End":"07:02.420","Text":"Well, that equals to 180 minus 175 divided by 2."},{"Start":"07:02.420 ","End":"07:04.190","Text":"Well, we\u0027ve calculated that,"},{"Start":"07:04.190 ","End":"07:07.755","Text":"that equals to 177.5."},{"Start":"07:07.755 ","End":"07:10.965","Text":"There we have it, that\u0027s the mode,"},{"Start":"07:10.965 ","End":"07:13.205","Text":"this is the median,"},{"Start":"07:13.205 ","End":"07:16.710","Text":"and this is the average."}],"ID":13317},{"Watched":false,"Name":"Exercise 9","Duration":"10m 18s","ChapterTopicVideoID":12834,"CourseChapterTopicPlaylistID":265041,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12834.jpeg","UploadDate":"2018-07-30T17:29:49.2970000","DurationForVideoObject":"PT10M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this question, we have students who are working in"},{"Start":"00:02.940 ","End":"00:07.305","Text":"a university faculty and we\u0027re analyzing their hourly wages."},{"Start":"00:07.305 ","End":"00:10.575","Text":"The results are displayed in a histogram."},{"Start":"00:10.575 ","End":"00:13.170","Text":"If we recall on a histogram,"},{"Start":"00:13.170 ","End":"00:16.875","Text":"the x-axis, okay, is the variable."},{"Start":"00:16.875 ","End":"00:21.480","Text":"The variable here is the hourly wages and that\u0027s in dollars per hour,"},{"Start":"00:21.480 ","End":"00:26.435","Text":"that\u0027s the units and on the y-axis where we have the density."},{"Start":"00:26.435 ","End":"00:31.025","Text":"Now we\u0027re asked to find the mode of the probability distribution."},{"Start":"00:31.025 ","End":"00:34.700","Text":"Now, how do we calculate the mode in a histogram?"},{"Start":"00:34.700 ","End":"00:36.710","Text":"Well, if we recall,"},{"Start":"00:36.710 ","End":"00:41.660","Text":"the mode is defined as the x value that\u0027s"},{"Start":"00:41.660 ","End":"00:47.930","Text":"the midpoint of the category that has the highest density."},{"Start":"00:47.930 ","End":"00:54.860","Text":"Here then we can see that this is the highest density and it\u0027s in the first category."},{"Start":"00:54.860 ","End":"00:57.080","Text":"What\u0027s the midpoint right here?"},{"Start":"00:57.080 ","End":"00:59.765","Text":"Well, the boundaries here are between 20 and 30."},{"Start":"00:59.765 ","End":"01:02.300","Text":"The midpoint is 25,"},{"Start":"01:02.300 ","End":"01:08.695","Text":"that means that the mode here is $25 per hour."},{"Start":"01:08.695 ","End":"01:13.020","Text":"In this section, we\u0027re asked to find the median of the probability distribution."},{"Start":"01:13.020 ","End":"01:15.005","Text":"Well again, here\u0027s our histogram,"},{"Start":"01:15.005 ","End":"01:17.060","Text":"and here\u0027s a density right here."},{"Start":"01:17.060 ","End":"01:19.670","Text":"That\u0027s the y-axis and that\u0027s the x-axis,"},{"Start":"01:19.670 ","End":"01:23.020","Text":"that\u0027s the rate, that\u0027s the dollars per hour."},{"Start":"01:23.020 ","End":"01:25.670","Text":"Now, whenever we have a histogram,"},{"Start":"01:25.670 ","End":"01:28.370","Text":"it\u0027s pretty difficult to calculate the median."},{"Start":"01:28.370 ","End":"01:31.430","Text":"If you recall, I suggested that we convert"},{"Start":"01:31.430 ","End":"01:37.150","Text":"the histogram into a frequency table by categories. Let\u0027s do that."},{"Start":"01:37.150 ","End":"01:40.230","Text":"Here is a frequency table right here."},{"Start":"01:40.230 ","End":"01:44.630","Text":"Here we have our x values and categories, right?"},{"Start":"01:44.630 ","End":"01:45.950","Text":"We have 20-30,"},{"Start":"01:45.950 ","End":"01:47.090","Text":"that\u0027s the first category,"},{"Start":"01:47.090 ","End":"01:49.280","Text":"that\u0027s right here, 30-40,"},{"Start":"01:49.280 ","End":"01:50.630","Text":"that\u0027s the second category,"},{"Start":"01:50.630 ","End":"01:52.195","Text":"and so on and so forth."},{"Start":"01:52.195 ","End":"01:57.079","Text":"Here we have the density and what\u0027s required here is the frequency."},{"Start":"01:57.079 ","End":"01:59.510","Text":"If we recall, the density,"},{"Start":"01:59.510 ","End":"02:04.985","Text":"well that equals to the frequency divided by L_1 minus L_0."},{"Start":"02:04.985 ","End":"02:11.095","Text":"That\u0027s the frequency of the category divided by the width of the category,"},{"Start":"02:11.095 ","End":"02:13.280","Text":"but the frequencies, what we\u0027re looking for,"},{"Start":"02:13.280 ","End":"02:15.800","Text":"we have the width and we have the density."},{"Start":"02:15.800 ","End":"02:18.440","Text":"That means that the frequency,"},{"Start":"02:18.440 ","End":"02:24.405","Text":"well that equals to the density times the width."},{"Start":"02:24.405 ","End":"02:28.520","Text":"Let\u0027s just calculate the frequency for each 1 of the categories."},{"Start":"02:28.520 ","End":"02:30.755","Text":"For the first category, well,"},{"Start":"02:30.755 ","End":"02:35.990","Text":"the density that\u0027s 5 times, well, what\u0027s the width?"},{"Start":"02:35.990 ","End":"02:38.770","Text":"Well, that\u0027s 30 minus 20, that\u0027s 10."},{"Start":"02:38.770 ","End":"02:41.430","Text":"That equals to 50."},{"Start":"02:41.430 ","End":"02:42.855","Text":"Let\u0027s do another 1."},{"Start":"02:42.855 ","End":"02:47.730","Text":"The second category, the density is 4 and the width is 10."},{"Start":"02:47.730 ","End":"02:52.120","Text":"That means that the frequency is 40."},{"Start":"02:52.160 ","End":"02:56.795","Text":"Now, let\u0027s do all the rest. Here we go."},{"Start":"02:56.795 ","End":"03:01.565","Text":"Those are our frequencies for each 1 of the categories right here."},{"Start":"03:01.565 ","End":"03:04.060","Text":"What\u0027s our n value?"},{"Start":"03:04.060 ","End":"03:07.050","Text":"Well, that equals to 180."},{"Start":"03:07.050 ","End":"03:09.525","Text":"We just sum up these values."},{"Start":"03:09.525 ","End":"03:17.180","Text":"How do we calculate the median in a frequency table when we have categories?"},{"Start":"03:17.180 ","End":"03:23.570","Text":"Well, the first stage is to see in which category the median lies and secondly,"},{"Start":"03:23.570 ","End":"03:25.730","Text":"we\u0027ll use our formula."},{"Start":"03:25.730 ","End":"03:27.905","Text":"If for the first stage,"},{"Start":"03:27.905 ","End":"03:31.885","Text":"we have to take n and we have to divide it by 2."},{"Start":"03:31.885 ","End":"03:33.432","Text":"N divided by 2,"},{"Start":"03:33.432 ","End":"03:35.700","Text":"that means it\u0027s a 180 divided by 2,"},{"Start":"03:35.700 ","End":"03:37.830","Text":"that means that that\u0027s 90."},{"Start":"03:37.830 ","End":"03:43.020","Text":"We\u0027re looking for the x value in the 90th place."},{"Start":"03:43.020 ","End":"03:44.585","Text":"Now, in order to do that,"},{"Start":"03:44.585 ","End":"03:47.380","Text":"we\u0027ll need the cumulative frequencies."},{"Start":"03:47.380 ","End":"03:49.125","Text":"Let\u0027s just calculate that."},{"Start":"03:49.125 ","End":"03:51.030","Text":"Well, that\u0027s 50 right here,"},{"Start":"03:51.030 ","End":"03:53.190","Text":"50 plus 40, that\u0027s 90."},{"Start":"03:53.190 ","End":"03:55.545","Text":"90 plus 30, that\u0027s 120."},{"Start":"03:55.545 ","End":"03:57.090","Text":"120 plus 20,"},{"Start":"03:57.090 ","End":"04:01.650","Text":"that\u0027s 140, and 140 plus 40, that\u0027s 180."},{"Start":"04:01.650 ","End":"04:09.020","Text":"We can see that since we\u0027re looking for the x value in the 90th place,"},{"Start":"04:09.020 ","End":"04:11.120","Text":"well here in our cumulative frequency,"},{"Start":"04:11.120 ","End":"04:17.570","Text":"we have the 90th place as the top value for this category."},{"Start":"04:17.570 ","End":"04:23.330","Text":"We can automatically see that our median is equal to 40,"},{"Start":"04:23.330 ","End":"04:26.570","Text":"but let\u0027s just calculate"},{"Start":"04:26.570 ","End":"04:31.010","Text":"the median using the standard formula to make sure that we\u0027re right."},{"Start":"04:31.010 ","End":"04:38.510","Text":"The median, well that equals to L_0 plus n divided by 2"},{"Start":"04:38.510 ","End":"04:45.080","Text":"minus cumulative distribution of X_m minus 1"},{"Start":"04:45.080 ","End":"04:53.130","Text":"divided by the frequency of X_m times L_1 minus L_0."},{"Start":"04:53.360 ","End":"04:55.440","Text":"What does that equal to?"},{"Start":"04:55.440 ","End":"05:03.175","Text":"Well, L_0, that\u0027s the lower bound of the category that has the median,"},{"Start":"05:03.175 ","End":"05:08.330","Text":"that\u0027s 30 plus n divided by 2, that\u0027s 90."},{"Start":"05:08.330 ","End":"05:10.450","Text":"We\u0027ve calculated that right here."},{"Start":"05:10.450 ","End":"05:13.160","Text":"90 minus, what\u0027s"},{"Start":"05:13.160 ","End":"05:20.375","Text":"the cumulative frequency of the category preceding the category that has the median?"},{"Start":"05:20.375 ","End":"05:23.555","Text":"That\u0027s this guy right here, that\u0027s 50,"},{"Start":"05:23.555 ","End":"05:29.345","Text":"divided by the frequency of the category that has the median,"},{"Start":"05:29.345 ","End":"05:35.460","Text":"that\u0027s 40 times L_1 minus L_0."},{"Start":"05:35.460 ","End":"05:39.020","Text":"That\u0027s the width of the category that has the median."},{"Start":"05:39.020 ","End":"05:43.175","Text":"That\u0027s 40 minus 30 and that\u0027s 10."},{"Start":"05:43.175 ","End":"05:45.955","Text":"That\u0027s 40 minus 30."},{"Start":"05:45.955 ","End":"05:48.615","Text":"That comes out to what?"},{"Start":"05:48.615 ","End":"05:50.700","Text":"90 minus 50 that\u0027s 40,"},{"Start":"05:50.700 ","End":"05:52.110","Text":"40 divided by 40,"},{"Start":"05:52.110 ","End":"05:54.660","Text":"that\u0027s 1 times 10, that\u0027s 10."},{"Start":"05:54.660 ","End":"05:57.555","Text":"So 30 plus 10, that equals to 40."},{"Start":"05:57.555 ","End":"06:01.580","Text":"Here we calculated the median in 2 different ways,"},{"Start":"06:01.580 ","End":"06:06.380","Text":"1 is through the classical formula and the other just by using logic,"},{"Start":"06:06.380 ","End":"06:12.870","Text":"just by seeing that the cumulative frequency here in the second category, that\u0027s 90."},{"Start":"06:12.870 ","End":"06:18.065","Text":"That means, that since the median is in the 90th place,"},{"Start":"06:18.065 ","End":"06:26.220","Text":"then the median has to be the upper bound of this category and that\u0027s 40."},{"Start":"06:26.740 ","End":"06:30.920","Text":"In this section, we\u0027re asked to explain without doing any calculations,"},{"Start":"06:30.920 ","End":"06:35.090","Text":"whether the average is larger or smaller or equal to the median."},{"Start":"06:35.090 ","End":"06:39.530","Text":"Let\u0027s take a look at the shape of this distribution right here."},{"Start":"06:39.530 ","End":"06:41.875","Text":"It looks something like this."},{"Start":"06:41.875 ","End":"06:46.150","Text":"This is a right asymmetrical distribution."},{"Start":"06:46.150 ","End":"06:48.680","Text":"If we have our x-axis,"},{"Start":"06:48.680 ","End":"06:51.845","Text":"we see that in this type of distribution,"},{"Start":"06:51.845 ","End":"06:55.730","Text":"a large number of the observations have low values for"},{"Start":"06:55.730 ","End":"07:00.230","Text":"X and a small number of observations have large values of X."},{"Start":"07:00.230 ","End":"07:05.795","Text":"So how does that impact the measures of centrality here?"},{"Start":"07:05.795 ","End":"07:08.825","Text":"If we recall what we said about"},{"Start":"07:08.825 ","End":"07:12.865","Text":"right asymmetric distributions with regard to the metrics."},{"Start":"07:12.865 ","End":"07:14.900","Text":"First comes the mode,"},{"Start":"07:14.900 ","End":"07:17.015","Text":"then comes the median,"},{"Start":"07:17.015 ","End":"07:20.450","Text":"then the average, and then the mid-range."},{"Start":"07:20.450 ","End":"07:27.110","Text":"That means that for all distributions with the right asymmetry,"},{"Start":"07:27.110 ","End":"07:32.340","Text":"the average will always be greater than the median."},{"Start":"07:32.590 ","End":"07:36.200","Text":"In this section, we learned that a number of students into"},{"Start":"07:36.200 ","End":"07:40.175","Text":"$20-30 category must be removed from the population."},{"Start":"07:40.175 ","End":"07:42.380","Text":"We\u0027re asked how will this affect the average,"},{"Start":"07:42.380 ","End":"07:43.550","Text":"the median, and the mode?"},{"Start":"07:43.550 ","End":"07:47.435","Text":"We are asked to explain the answer without doing any calculations."},{"Start":"07:47.435 ","End":"07:49.640","Text":"Okay, this is our histogram."},{"Start":"07:49.640 ","End":"07:51.890","Text":"This is the first category,"},{"Start":"07:51.890 ","End":"07:56.584","Text":"the second category, the third category, the fourth category,"},{"Start":"07:56.584 ","End":"07:58.145","Text":"and the fifth category,"},{"Start":"07:58.145 ","End":"08:01.490","Text":"and we\u0027re told that we are taking some of"},{"Start":"08:01.490 ","End":"08:06.180","Text":"the students in this category and we\u0027re removing them."},{"Start":"08:06.320 ","End":"08:08.595","Text":"Now, having said that,"},{"Start":"08:08.595 ","End":"08:11.475","Text":"what happens to the mode?"},{"Start":"08:11.475 ","End":"08:15.290","Text":"When we remove the students from this category,"},{"Start":"08:15.290 ","End":"08:19.445","Text":"then the density has to decrease."},{"Start":"08:19.445 ","End":"08:24.875","Text":"It can decrease in such a way that it\u0027ll still be higher,"},{"Start":"08:24.875 ","End":"08:28.880","Text":"that the density would still be higher than the density in the second category."},{"Start":"08:28.880 ","End":"08:30.200","Text":"If that\u0027s the case,"},{"Start":"08:30.200 ","End":"08:34.240","Text":"then the mode will not change,"},{"Start":"08:34.240 ","End":"08:37.260","Text":"and on the other hand,"},{"Start":"08:37.260 ","End":"08:43.460","Text":"if it decreases to a level below the density of the second category,"},{"Start":"08:43.460 ","End":"08:46.565","Text":"then the mode will change."},{"Start":"08:46.565 ","End":"08:55.080","Text":"Just remember the mode is the midpoint of the category with the highest density."},{"Start":"08:55.080 ","End":"08:58.265","Text":"Again, if after the change,"},{"Start":"08:58.265 ","End":"09:01.775","Text":"the density is still higher than this category right here,"},{"Start":"09:01.775 ","End":"09:03.560","Text":"then the mode will not change,"},{"Start":"09:03.560 ","End":"09:08.180","Text":"but if it falls below the density in the second category,"},{"Start":"09:08.180 ","End":"09:11.620","Text":"well then the mode will be here."},{"Start":"09:11.620 ","End":"09:14.775","Text":"Okay, what about the median?"},{"Start":"09:14.775 ","End":"09:17.480","Text":"We know that if this is the mode,"},{"Start":"09:17.480 ","End":"09:20.060","Text":"the median is greater than the mode."},{"Start":"09:20.060 ","End":"09:22.475","Text":"What\u0027s the definition of the median?"},{"Start":"09:22.475 ","End":"09:28.220","Text":"The median says that 50 percent of the population is below this value and"},{"Start":"09:28.220 ","End":"09:34.130","Text":"50 percent is above this value and if we take away some of the students in this category,"},{"Start":"09:34.130 ","End":"09:39.499","Text":"that means that here on the left side of the median,"},{"Start":"09:39.499 ","End":"09:42.560","Text":"we have now less than 50 percent."},{"Start":"09:42.560 ","End":"09:46.670","Text":"In order to get our 50 percent balance right,"},{"Start":"09:46.670 ","End":"09:52.545","Text":"then the median has to increase."},{"Start":"09:52.545 ","End":"09:55.025","Text":"What about the average?"},{"Start":"09:55.025 ","End":"09:58.265","Text":"Well, we know that the average is greater than the median."},{"Start":"09:58.265 ","End":"10:04.070","Text":"If we remove students that have data which is"},{"Start":"10:04.070 ","End":"10:10.609","Text":"less than the average in this category is obviously less than the average,"},{"Start":"10:10.609 ","End":"10:17.370","Text":"then that means that the average must increase as well."}],"ID":13318}],"Thumbnail":null,"ID":265041},{"Name":"Measures of Dispersion - Range, Variance, and Standard Deviation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"3m ","ChapterTopicVideoID":12840,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12840.jpeg","UploadDate":"2018-07-30T17:49:49.1970000","DurationForVideoObject":"PT3M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this chapter we\u0027ll be talking about Descriptive Statistics,"},{"Start":"00:02.760 ","End":"00:05.040","Text":"specifically Measures of Dispersion."},{"Start":"00:05.040 ","End":"00:06.540","Text":"We\u0027ll be talking about the Range,"},{"Start":"00:06.540 ","End":"00:09.375","Text":"the Variance, and the Standard Deviation."},{"Start":"00:09.375 ","End":"00:13.845","Text":"Now the goal of these measures is to measure the dispersion of the data."},{"Start":"00:13.845 ","End":"00:18.705","Text":"That means how distant and different are the observations from each other."},{"Start":"00:18.705 ","End":"00:21.435","Text":"Now, when we first looked at data,"},{"Start":"00:21.435 ","End":"00:23.910","Text":"we learned how to present the data,"},{"Start":"00:23.910 ","End":"00:26.790","Text":"how to describe the data in chart form,"},{"Start":"00:26.790 ","End":"00:29.535","Text":"in table form, in graph form."},{"Start":"00:29.535 ","End":"00:34.020","Text":"The second step was to analyze the measure of central tendency."},{"Start":"00:34.020 ","End":"00:37.255","Text":"We want to know what is the center of the distribution."},{"Start":"00:37.255 ","End":"00:39.845","Text":"We looked at the mode, the mean,"},{"Start":"00:39.845 ","End":"00:42.575","Text":"and the median, but that\u0027s not enough."},{"Start":"00:42.575 ","End":"00:47.390","Text":"We also need to look at the dispersion of the data."},{"Start":"00:47.390 ","End":"00:49.865","Text":"Now, why is that?"},{"Start":"00:49.865 ","End":"00:55.059","Text":"Let\u0027s take an example of marks in a class."},{"Start":"00:55.059 ","End":"00:57.530","Text":"Here we have a table,"},{"Start":"00:57.530 ","End":"00:59.705","Text":"a frequency table of marks."},{"Start":"00:59.705 ","End":"01:02.315","Text":"Here in the first column we have the marks,"},{"Start":"01:02.315 ","End":"01:06.050","Text":"in the second column we have the frequency of students receiving these marks."},{"Start":"01:06.050 ","End":"01:09.050","Text":"For example, 2 students received the mark of 5,"},{"Start":"01:09.050 ","End":"01:12.125","Text":"4 students a mark of 6 and so on and so forth."},{"Start":"01:12.125 ","End":"01:15.410","Text":"Now, what we\u0027re interested in is to"},{"Start":"01:15.410 ","End":"01:19.765","Text":"see whether all the students have the same level of understanding."},{"Start":"01:19.765 ","End":"01:23.075","Text":"If they do have the same level of understanding,"},{"Start":"01:23.075 ","End":"01:25.880","Text":"then they\u0027d get pretty much the same mark."},{"Start":"01:25.880 ","End":"01:29.285","Text":"But if they don\u0027t have the same level of understanding,"},{"Start":"01:29.285 ","End":"01:33.860","Text":"then they\u0027d be all over the place with respect to the marks."},{"Start":"01:33.860 ","End":"01:37.625","Text":"Now, let\u0027s talk about stocks in the stock exchange."},{"Start":"01:37.625 ","End":"01:40.730","Text":"Now, we want to know about the risk of a stock."},{"Start":"01:40.730 ","End":"01:46.100","Text":"It\u0027s not enough to know the average daily return or monthly return of a stock."},{"Start":"01:46.100 ","End":"01:48.740","Text":"We want to know the risks involved,"},{"Start":"01:48.740 ","End":"01:52.150","Text":"the fluctuations of the stock itself."},{"Start":"01:52.150 ","End":"01:56.615","Text":"This is why we talk about measures of dispersion."},{"Start":"01:56.615 ","End":"01:59.495","Text":"Now the first measure is the range."},{"Start":"01:59.495 ","End":"02:02.060","Text":"Now, the range is defined as"},{"Start":"02:02.060 ","End":"02:06.725","Text":"the difference between the highest and the lowest observation."},{"Start":"02:06.725 ","End":"02:14.970","Text":"Here for example, we\u0027re looking at the maximum value of x, that\u0027s x_max."},{"Start":"02:14.970 ","End":"02:18.590","Text":"Here in our case that will be equal to 10."},{"Start":"02:18.590 ","End":"02:21.500","Text":"What about the minimum value of x?"},{"Start":"02:21.500 ","End":"02:23.165","Text":"That\u0027s x_min."},{"Start":"02:23.165 ","End":"02:25.990","Text":"That equals to 5."},{"Start":"02:25.990 ","End":"02:29.450","Text":"The range would be the maximum value,"},{"Start":"02:29.450 ","End":"02:33.155","Text":"that\u0027s 10 minus the minimum value that\u0027s 5."},{"Start":"02:33.155 ","End":"02:36.395","Text":"The range here would be 5."},{"Start":"02:36.395 ","End":"02:43.415","Text":"Now, this isn\u0027t such a good measure because it doesn\u0027t take into account the frequency."},{"Start":"02:43.415 ","End":"02:45.920","Text":"It just takes into account the values,"},{"Start":"02:45.920 ","End":"02:50.585","Text":"the highest and lowest value of x."},{"Start":"02:50.585 ","End":"02:52.805","Text":"This isn\u0027t a good measure."},{"Start":"02:52.805 ","End":"02:55.985","Text":"We\u0027re not going to rely on it that much."},{"Start":"02:55.985 ","End":"03:01.560","Text":"We\u0027ll be relying more on the variance and standard deviation."}],"ID":13319},{"Watched":false,"Name":"Tutorial 2","Duration":"6m 36s","ChapterTopicVideoID":12841,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12841.jpeg","UploadDate":"2018-07-30T17:50:22.2970000","DurationForVideoObject":"PT6M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.365","Text":"The most common Measures of Dispersion are the Variance and the Standard Deviation."},{"Start":"00:04.365 ","End":"00:09.390","Text":"Now, the variance is defined as the average of the squares of the deviations."},{"Start":"00:09.390 ","End":"00:10.875","Text":"Now, what\u0027s a deviation?"},{"Start":"00:10.875 ","End":"00:15.285","Text":"The deviation is the distance of the observation from the average,"},{"Start":"00:15.285 ","End":"00:18.570","Text":"and the standard deviation is the square root of the variance."},{"Start":"00:18.570 ","End":"00:20.070","Text":"Now, from this sense,"},{"Start":"00:20.070 ","End":"00:21.960","Text":"you\u0027re not going to understand the variances."},{"Start":"00:21.960 ","End":"00:23.745","Text":"Let\u0027s take a look at an example."},{"Start":"00:23.745 ","End":"00:29.340","Text":"Here we\u0027re asked to calculate the variance of the following number series 5, 4, and 9."},{"Start":"00:29.340 ","End":"00:35.260","Text":"The first thing that I want to do is I want to count how many observations that I have."},{"Start":"00:35.260 ","End":"00:37.680","Text":"N here is equal to 3,"},{"Start":"00:37.680 ","End":"00:39.405","Text":"I have 3 observations."},{"Start":"00:39.405 ","End":"00:43.040","Text":"The second thing that I want to do is I want to calculate the average."},{"Start":"00:43.040 ","End":"00:47.840","Text":"Well, that equals to the sum of my observations divided by n. That equals"},{"Start":"00:47.840 ","End":"00:53.599","Text":"to 5 plus 4 plus 9 divided by 3,"},{"Start":"00:53.599 ","End":"00:56.400","Text":"and that equals to 6."},{"Start":"00:56.630 ","End":"01:01.815","Text":"Now, I can calculate my variance."},{"Start":"01:01.815 ","End":"01:03.480","Text":"How do I do that?"},{"Start":"01:03.480 ","End":"01:06.710","Text":"I use this expression right here."},{"Start":"01:06.710 ","End":"01:11.240","Text":"This is a mathematical definition of the variance and what do we have here?"},{"Start":"01:11.240 ","End":"01:13.105","Text":"Let\u0027s just take a look at it."},{"Start":"01:13.105 ","End":"01:20.885","Text":"First of all, we have our observation XI and we subtract from that the average x-bar."},{"Start":"01:20.885 ","End":"01:25.640","Text":"Now, that\u0027s the deviation here and now we take this deviation,"},{"Start":"01:25.640 ","End":"01:28.880","Text":"we square that, so we have a deviation squared."},{"Start":"01:28.880 ","End":"01:32.180","Text":"We sum all these deviations squared,"},{"Start":"01:32.180 ","End":"01:35.000","Text":"and then we divide that by n,"},{"Start":"01:35.000 ","End":"01:36.920","Text":"by the number of observations."},{"Start":"01:36.920 ","End":"01:42.175","Text":"Then this would be the average of the deviations square."},{"Start":"01:42.175 ","End":"01:48.565","Text":"Let\u0027s just calculate the variance using this formula for these series."},{"Start":"01:48.565 ","End":"01:53.360","Text":"The first thing that I want to do is I want to calculate the deviations right here."},{"Start":"01:53.360 ","End":"01:57.155","Text":"That\u0027ll be x_i minus x bar."},{"Start":"01:57.155 ","End":"02:03.150","Text":"X1 would be 5 minus x bar with that 6,"},{"Start":"02:03.150 ","End":"02:06.520","Text":"which equals to 5 minus x, that\u0027s minus 1."},{"Start":"02:06.520 ","End":"02:14.270","Text":"Then we have 4 minus 6 that equals to minus 2 and 9 minus 6 that equals to 3."},{"Start":"02:14.270 ","End":"02:16.610","Text":"Now, if we sum these guys up,"},{"Start":"02:16.610 ","End":"02:18.050","Text":"that equal to 0,"},{"Start":"02:18.050 ","End":"02:19.670","Text":"they will sum up to 0."},{"Start":"02:19.670 ","End":"02:26.824","Text":"This is the first thing that we have to know that if we sum the deviations only,"},{"Start":"02:26.824 ","End":"02:29.510","Text":"it will always be equal to 0."},{"Start":"02:29.510 ","End":"02:30.830","Text":"Not only in this case,"},{"Start":"02:30.830 ","End":"02:33.875","Text":"but in all other cases, that\u0027s always."},{"Start":"02:33.875 ","End":"02:39.775","Text":"This didn\u0027t get as far with respect to what\u0027s the dispersion of this data."},{"Start":"02:39.775 ","End":"02:45.780","Text":"What we need to do is we need to take the deviations x_i minus x bar,"},{"Start":"02:45.780 ","End":"02:48.235","Text":"but now we want to square them."},{"Start":"02:48.235 ","End":"02:53.540","Text":"Now, minus 1 becomes 1, minus 1 squared,"},{"Start":"02:53.540 ","End":"02:54.991","Text":"minus 2 squared,"},{"Start":"02:54.991 ","End":"02:59.660","Text":"that becomes 4 and 3 squared, that becomes 9."},{"Start":"02:59.660 ","End":"03:02.165","Text":"1 plus 4 plus 9,"},{"Start":"03:02.165 ","End":"03:04.950","Text":"that is equals to 14."},{"Start":"03:05.750 ","End":"03:08.930","Text":"How do we calculate the variance?"},{"Start":"03:08.930 ","End":"03:12.965","Text":"That\u0027s this guy right here. Let\u0027s do that."},{"Start":"03:12.965 ","End":"03:18.140","Text":"That\u0027s the sum of the square of the deviations,"},{"Start":"03:18.140 ","End":"03:20.690","Text":"well these are the square of the deviations right here."},{"Start":"03:20.690 ","End":"03:22.715","Text":"All we have to do is sum them up."},{"Start":"03:22.715 ","End":"03:29.555","Text":"That will be S squared x that equals to 1 plus 4 plus 9"},{"Start":"03:29.555 ","End":"03:36.950","Text":"divided by n divided by 3 and that equals to 14 divided by 3,"},{"Start":"03:36.950 ","End":"03:40.415","Text":"and that\u0027s 4 and 2/3."},{"Start":"03:40.415 ","End":"03:44.510","Text":"Now, this is in units squared. What does that mean?"},{"Start":"03:44.510 ","End":"03:52.740","Text":"This number shows me how far away are the observations from the average in unit squared."},{"Start":"03:52.850 ","End":"03:55.654","Text":"This is very hard to understand."},{"Start":"03:55.654 ","End":"03:59.405","Text":"It\u0027s not very intuitive and that\u0027s why we have the standard deviation,"},{"Start":"03:59.405 ","End":"04:01.720","Text":"we\u0027ll get to that in a bit."},{"Start":"04:01.720 ","End":"04:06.380","Text":"There are several things that I want to say about the variance."},{"Start":"04:06.380 ","End":"04:10.040","Text":"The first thing is that we sometimes"},{"Start":"04:10.040 ","End":"04:16.730","Text":"see the variance written as sigma squared of x."},{"Start":"04:16.730 ","End":"04:19.970","Text":"That\u0027s the small Greek letter Sigma."},{"Start":"04:19.970 ","End":"04:26.735","Text":"Now it\u0027s not that common descriptive statistics to see this way of writing the variance,"},{"Start":"04:26.735 ","End":"04:28.640","Text":"but we do see it from time to time,"},{"Start":"04:28.640 ","End":"04:30.440","Text":"so don\u0027t be surprised."},{"Start":"04:30.440 ","End":"04:35.120","Text":"The second thing is that this expression right"},{"Start":"04:35.120 ","End":"04:42.230","Text":"here that equals to this expression the definition of the variance."},{"Start":"04:42.230 ","End":"04:44.960","Text":"By doing a lot of algebraic manipulation,"},{"Start":"04:44.960 ","End":"04:46.265","Text":"I won\u0027t get into that."},{"Start":"04:46.265 ","End":"04:50.735","Text":"This expression becomes this thing right here."},{"Start":"04:50.735 ","End":"04:57.650","Text":"Now, this is a much easier expression to work with when calculating the variance,"},{"Start":"04:57.650 ","End":"05:01.330","Text":"and I\u0027ll be using that instead of this in most cases."},{"Start":"05:01.330 ","End":"05:08.570","Text":"Let\u0027s just see how we can calculate the variance using this expression right here."},{"Start":"05:08.570 ","End":"05:10.775","Text":"What do we have to do?"},{"Start":"05:10.775 ","End":"05:12.740","Text":"We have to take observations,"},{"Start":"05:12.740 ","End":"05:14.710","Text":"we have to square them,"},{"Start":"05:14.710 ","End":"05:17.540","Text":"we have to sum the squares of the observations,"},{"Start":"05:17.540 ","End":"05:18.940","Text":"divide that by n,"},{"Start":"05:18.940 ","End":"05:24.505","Text":"and then we have to take and subtraction that the average squared."},{"Start":"05:24.505 ","End":"05:27.305","Text":"By using this expression,"},{"Start":"05:27.305 ","End":"05:30.020","Text":"S squared of x, what do we have here?"},{"Start":"05:30.020 ","End":"05:36.110","Text":"We have 5 squared and that\u0027s the first observation squared."},{"Start":"05:36.110 ","End":"05:44.330","Text":"Plus 4 squared plus 9 squared divided by n. N equals to 3 minus,"},{"Start":"05:44.330 ","End":"05:46.965","Text":"let\u0027s not forget that, the average squared,"},{"Start":"05:46.965 ","End":"05:52.375","Text":"that\u0027s 6 squared, and then also comes out to 4 and 2/3."},{"Start":"05:52.375 ","End":"05:55.730","Text":"Let\u0027s just go over the steps again."},{"Start":"05:55.730 ","End":"05:59.600","Text":"The first thing that we want to do is we want to calculate"},{"Start":"05:59.600 ","End":"06:04.530","Text":"the number of observations that we have,"},{"Start":"06:04.580 ","End":"06:07.560","Text":"this thing right here."},{"Start":"06:07.560 ","End":"06:12.320","Text":"Second, we want to calculate the average."},{"Start":"06:12.320 ","End":"06:17.615","Text":"The third thing that we want to do is we want to calculate the deviations."},{"Start":"06:17.615 ","End":"06:21.155","Text":"Fourth that we want to do is calculate the deviation"},{"Start":"06:21.155 ","End":"06:24.950","Text":"squared and the fifth and last thing that we"},{"Start":"06:24.950 ","End":"06:32.330","Text":"want to do is to plug in the numbers to calculate and calculate the variance."},{"Start":"06:32.330 ","End":"06:37.080","Text":"This is the method of calculating the variance."}],"ID":13320},{"Watched":false,"Name":"Tutorial 3","Duration":"5m 7s","ChapterTopicVideoID":12842,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12842.jpeg","UploadDate":"2018-07-30T17:51:25.0800000","DurationForVideoObject":"PT5M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.445","Text":"What happens if I have a frequency table,"},{"Start":"00:02.445 ","End":"00:04.440","Text":"how do I calculate the variance?"},{"Start":"00:04.440 ","End":"00:09.705","Text":"Well, I use the same formulas as I did in the previous section,"},{"Start":"00:09.705 ","End":"00:12.510","Text":"but, right now I have to take into"},{"Start":"00:12.510 ","End":"00:16.275","Text":"consideration the frequencies of the categories right here."},{"Start":"00:16.275 ","End":"00:20.940","Text":"Let\u0027s say I have a frequency table like this."},{"Start":"00:20.940 ","End":"00:22.590","Text":"I have here the marks,"},{"Start":"00:22.590 ","End":"00:23.820","Text":"let\u0027s say of students,"},{"Start":"00:23.820 ","End":"00:26.580","Text":"and I want to know what the frequency of the students are,"},{"Start":"00:26.580 ","End":"00:28.015","Text":"that get different marks."},{"Start":"00:28.015 ","End":"00:30.690","Text":"I have 2 students who get a mark of 5,"},{"Start":"00:30.690 ","End":"00:34.175","Text":"4 students who get a mark of 6 and so on and so forth."},{"Start":"00:34.175 ","End":"00:38.780","Text":"Now, the first thing that I want to do is I want to see how many students do I have?"},{"Start":"00:38.780 ","End":"00:40.775","Text":"I have 25 students here,"},{"Start":"00:40.775 ","End":"00:43.385","Text":"n here equals to 25."},{"Start":"00:43.385 ","End":"00:45.590","Text":"Let\u0027s just write that down here."},{"Start":"00:45.590 ","End":"00:47.615","Text":"What about the average?"},{"Start":"00:47.615 ","End":"00:52.025","Text":"Well, the average is given to us, that\u0027s 7.44."},{"Start":"00:52.025 ","End":"00:56.390","Text":"Now, I can calculate the variance."},{"Start":"00:56.390 ","End":"00:59.239","Text":"Now I won\u0027t be using this formula right here."},{"Start":"00:59.239 ","End":"01:04.445","Text":"We said that this formula is much easier to work with."},{"Start":"01:04.445 ","End":"01:06.215","Text":"What do I need to do?"},{"Start":"01:06.215 ","End":"01:12.760","Text":"Well, I need to sum all the observation squared times the frequency,"},{"Start":"01:12.760 ","End":"01:17.365","Text":"divide that by n, and then subtract from that the average squared."},{"Start":"01:17.365 ","End":"01:20.760","Text":"Let\u0027s do that, then be s squared of x,"},{"Start":"01:20.760 ","End":"01:22.200","Text":"that\u0027s the variance of x."},{"Start":"01:22.200 ","End":"01:23.865","Text":"Now, that equals to what?"},{"Start":"01:23.865 ","End":"01:28.457","Text":"Well, x_1 squared, well that\u0027s 5 squared,"},{"Start":"01:28.457 ","End":"01:30.550","Text":"5 squared times the frequency,"},{"Start":"01:30.550 ","End":"01:32.275","Text":"the frequency is 2."},{"Start":"01:32.275 ","End":"01:34.840","Text":"Plus, we\u0027re summing up everything,"},{"Start":"01:34.840 ","End":"01:38.930","Text":"6 squared times 4"},{"Start":"01:38.930 ","End":"01:46.795","Text":"plus 7 squared times 8 and so on and so forth."},{"Start":"01:46.795 ","End":"01:56.660","Text":"All this sum, I have to divide by n. I just created another column here,"},{"Start":"01:56.660 ","End":"01:59.060","Text":"which is x squared times f,"},{"Start":"01:59.060 ","End":"02:02.505","Text":"so 5 squared times 2,"},{"Start":"02:02.505 ","End":"02:05.205","Text":"that\u0027s 25 times 2, that\u0027s 50,"},{"Start":"02:05.205 ","End":"02:07.220","Text":"and 6 squared times 4,"},{"Start":"02:07.220 ","End":"02:09.050","Text":"that\u0027s 36 times 4,"},{"Start":"02:09.050 ","End":"02:12.695","Text":"and that equals to 144,"},{"Start":"02:12.695 ","End":"02:14.135","Text":"and so on and so forth."},{"Start":"02:14.135 ","End":"02:17.120","Text":"That means that this equals to,"},{"Start":"02:17.120 ","End":"02:21.763","Text":"in the numerator I have 1,430,"},{"Start":"02:21.763 ","End":"02:23.435","Text":"and in the denominator,"},{"Start":"02:23.435 ","End":"02:25.010","Text":"I have n,"},{"Start":"02:25.010 ","End":"02:27.035","Text":"n here is 25."},{"Start":"02:27.035 ","End":"02:30.740","Text":"I can\u0027t forget this."},{"Start":"02:30.740 ","End":"02:35.330","Text":"I have to subtract from that x-bar squared, the average squared."},{"Start":"02:35.330 ","End":"02:40.145","Text":"That\u0027ll be minus 7.44 squared,"},{"Start":"02:40.145 ","End":"02:48.605","Text":"and that comes out to 1.8464."},{"Start":"02:48.605 ","End":"02:51.290","Text":"Now, this is exactly what we have here."},{"Start":"02:51.290 ","End":"02:58.090","Text":"We have the variance of x and that equals to the sum from i equaling 1"},{"Start":"02:58.090 ","End":"03:04.565","Text":"to n of x squared times the frequency divided by n minus x bar squared,"},{"Start":"03:04.565 ","End":"03:08.900","Text":"and it said, that equals to 1430 divided by 25"},{"Start":"03:08.900 ","End":"03:14.329","Text":"minus 7.44 squared and this then is the variance,"},{"Start":"03:14.329 ","End":"03:18.110","Text":"but this is in units squared,"},{"Start":"03:18.110 ","End":"03:19.985","Text":"this is in mark squared."},{"Start":"03:19.985 ","End":"03:25.305","Text":"This isn\u0027t very intuitive and that\u0027s why we have the standard deviation."},{"Start":"03:25.305 ","End":"03:27.800","Text":"The standard deviation, S,"},{"Start":"03:27.800 ","End":"03:32.485","Text":"that equals to the square root of the variance."},{"Start":"03:32.485 ","End":"03:35.390","Text":"That means that in our example,"},{"Start":"03:35.390 ","End":"03:39.695","Text":"that would be the square root of 1.8464,"},{"Start":"03:39.695 ","End":"03:44.560","Text":"and that equals to 1.3588."},{"Start":"03:44.560 ","End":"03:47.175","Text":"What\u0027s that?"},{"Start":"03:47.175 ","End":"03:57.307","Text":"This then is the average deviation of an observation from the average."},{"Start":"03:57.307 ","End":"04:02.270","Text":"This is in units of the grade that the student got."},{"Start":"04:02.270 ","End":"04:04.985","Text":"We can say that on average,"},{"Start":"04:04.985 ","End":"04:08.795","Text":"each student deviated from the average,"},{"Start":"04:08.795 ","End":"04:11.900","Text":"the average is this thing, 7.44."},{"Start":"04:11.900 ","End":"04:20.375","Text":"Each student deviated from this average by an average of 1.3588 marks."},{"Start":"04:20.375 ","End":"04:24.454","Text":"This then is the standard deviation."},{"Start":"04:24.454 ","End":"04:26.905","Text":"All it does as we said,"},{"Start":"04:26.905 ","End":"04:30.710","Text":"it doesn\u0027t give us the minimum or maximum value of x,"},{"Start":"04:30.710 ","End":"04:36.215","Text":"it doesn\u0027t point to where the majority of the students are, whether less students,"},{"Start":"04:36.215 ","End":"04:43.609","Text":"all it does, it gives us the average distance of an observation from the average."},{"Start":"04:43.609 ","End":"04:48.140","Text":"Now, when we have categories here,"},{"Start":"04:48.140 ","End":"04:52.897","Text":"instead of individual marks, well then,"},{"Start":"04:52.897 ","End":"04:57.890","Text":"we use the midpoint of the category for the x value."},{"Start":"04:57.890 ","End":"05:01.850","Text":"Then we use that to calculate the variance."},{"Start":"05:01.850 ","End":"05:04.325","Text":"There we have it,"},{"Start":"05:04.325 ","End":"05:08.040","Text":"let\u0027s get cracking and start doing exercises."}],"ID":13321},{"Watched":false,"Name":"Exercise 1","Duration":"4m 30s","ChapterTopicVideoID":12843,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12843.jpeg","UploadDate":"2018-07-30T17:51:38.1270000","DurationForVideoObject":"PT4M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this question, we\u0027re given the marks of"},{"Start":"00:02.040 ","End":"00:05.625","Text":"20 students completed a reading comprehension test."},{"Start":"00:05.625 ","End":"00:06.930","Text":"These are the marks right here."},{"Start":"00:06.930 ","End":"00:08.880","Text":"We\u0027re asked to calculate the variance,"},{"Start":"00:08.880 ","End":"00:10.080","Text":"and standard deviation,"},{"Start":"00:10.080 ","End":"00:12.495","Text":"and the range of the marks."},{"Start":"00:12.495 ","End":"00:19.290","Text":"In descriptive statistics, we are dealing with the analysis of groups."},{"Start":"00:19.290 ","End":"00:21.465","Text":"What\u0027s our group here?"},{"Start":"00:21.465 ","End":"00:27.435","Text":"Well, that\u0027s the 20 students that completed the comprehension tests."},{"Start":"00:27.435 ","End":"00:33.450","Text":"What\u0027s the variable by which we analyzed this group?"},{"Start":"00:33.450 ","End":"00:36.390","Text":"Well, that would be the marks."},{"Start":"00:36.390 ","End":"00:42.560","Text":"Let\u0027s start with the range of the marks."},{"Start":"00:42.560 ","End":"00:45.935","Text":"The range, which is denoted as R,"},{"Start":"00:45.935 ","End":"00:52.530","Text":"that equals to X_max minus X_min."},{"Start":"00:52.530 ","End":"00:56.915","Text":"If we take a look at this list of marks,"},{"Start":"00:56.915 ","End":"01:02.815","Text":"what\u0027s the maximum value that we have here? Well, that\u0027s 10."},{"Start":"01:02.815 ","End":"01:06.920","Text":"Lets write here. What\u0027s the minimum value?"},{"Start":"01:06.920 ","End":"01:10.900","Text":"Well, here, we have a 4."},{"Start":"01:10.900 ","End":"01:13.560","Text":"They\u0027ll be 10 minus 4,"},{"Start":"01:13.560 ","End":"01:15.435","Text":"and that\u0027ll be 6."},{"Start":"01:15.435 ","End":"01:20.800","Text":"The range here would be 6 marks."},{"Start":"01:20.930 ","End":"01:24.500","Text":"What about the variance and standard deviation?"},{"Start":"01:24.500 ","End":"01:27.530","Text":"Well, let\u0027s start with the variance."},{"Start":"01:27.530 ","End":"01:29.630","Text":"In order to calculate the variance,"},{"Start":"01:29.630 ","End":"01:35.205","Text":"the first thing that we need to do is we need to calculate the average."},{"Start":"01:35.205 ","End":"01:37.715","Text":"We have our n right here,"},{"Start":"01:37.715 ","End":"01:40.955","Text":"n here equals to 20, we\u0027re given that."},{"Start":"01:40.955 ","End":"01:42.500","Text":"But what about the average?"},{"Start":"01:42.500 ","End":"01:48.905","Text":"Well, the average is the sum of the observations divided by the number of observations."},{"Start":"01:48.905 ","End":"01:53.210","Text":"Well, that means that I have to add up all these observations."},{"Start":"01:53.210 ","End":"01:58.010","Text":"They\u0027ll be 7 plus 6 plus 8 plus 9 plus,"},{"Start":"01:58.010 ","End":"02:01.235","Text":"and so on and so forth for all 20 observations,"},{"Start":"02:01.235 ","End":"02:05.215","Text":"divided by n, where n here equals to 20."},{"Start":"02:05.215 ","End":"02:08.600","Text":"That means that the average here,"},{"Start":"02:08.600 ","End":"02:10.355","Text":"after doing the calculations,"},{"Start":"02:10.355 ","End":"02:13.915","Text":"that comes out to 6.9."},{"Start":"02:13.915 ","End":"02:18.185","Text":"We can start to calculate the variance."},{"Start":"02:18.185 ","End":"02:19.700","Text":"What\u0027s the variance?"},{"Start":"02:19.700 ","End":"02:22.795","Text":"Well, that\u0027s S squared of x."},{"Start":"02:22.795 ","End":"02:26.960","Text":"The definition of the variance is this."},{"Start":"02:26.960 ","End":"02:32.595","Text":"That\u0027s the sum of the deviation squared divided by"},{"Start":"02:32.595 ","End":"02:38.945","Text":"n. But what is our working expression here?"},{"Start":"02:38.945 ","End":"02:46.085","Text":"That\u0027s the sum of x squared divided by n minus the average squared."},{"Start":"02:46.085 ","End":"02:50.915","Text":"Let\u0027s use this thing because this is much easier to work with."},{"Start":"02:50.915 ","End":"03:00.590","Text":"That equals to 7 squared plus 6 squared plus 8 squared plus,"},{"Start":"03:00.590 ","End":"03:01.895","Text":"and so on and so forth."},{"Start":"03:01.895 ","End":"03:05.225","Text":"We\u0027re taking each observation, we\u0027re squaring it,"},{"Start":"03:05.225 ","End":"03:10.880","Text":"and then we\u0027re adding all of the squares of the observations divided by n,"},{"Start":"03:10.880 ","End":"03:12.830","Text":"n here is 20,"},{"Start":"03:12.830 ","End":"03:15.890","Text":"minus the average squared."},{"Start":"03:15.890 ","End":"03:17.390","Text":"Well, we calculated the average,"},{"Start":"03:17.390 ","End":"03:21.575","Text":"that\u0027s 6.9, that\u0027s 6.9 squared."},{"Start":"03:21.575 ","End":"03:25.520","Text":"That equals to 2.19,"},{"Start":"03:25.520 ","End":"03:26.990","Text":"and what\u0027s the units here?"},{"Start":"03:26.990 ","End":"03:30.735","Text":"Well, the units here are in marks squared."},{"Start":"03:30.735 ","End":"03:37.175","Text":"This isn\u0027t very intuitive and that\u0027s why we have the standard deviation."},{"Start":"03:37.175 ","End":"03:44.750","Text":"The standard deviation basically brings this value back to units that are intuitive."},{"Start":"03:44.750 ","End":"03:51.920","Text":"S here, that\u0027s the standard deviation that equals to the square root of the variance."},{"Start":"03:51.920 ","End":"04:00.865","Text":"That is the square root of 2.19 and that equals to 1.479,"},{"Start":"04:00.865 ","End":"04:03.840","Text":"and the units here are in marks."},{"Start":"04:03.840 ","End":"04:06.470","Text":"What does this value here mean?"},{"Start":"04:06.470 ","End":"04:10.250","Text":"Well, all it means is that each student has"},{"Start":"04:10.250 ","End":"04:14.165","Text":"a different grade after completing the comprehension tests."},{"Start":"04:14.165 ","End":"04:24.570","Text":"But on average, the distance of each student from the average is 1.479 marks."},{"Start":"04:24.970 ","End":"04:31.260","Text":"That\u0027s the intuitive meaning of the standard deviation in our case."}],"ID":13322},{"Watched":false,"Name":"Exercise 2","Duration":"6m 20s","ChapterTopicVideoID":12844,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12844.jpeg","UploadDate":"2018-07-30T17:52:01.6600000","DurationForVideoObject":"PT6M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this question, we\u0027re given a table that displays"},{"Start":"00:02.820 ","End":"00:06.105","Text":"the number of cars owned by a family in a certain community."},{"Start":"00:06.105 ","End":"00:08.970","Text":"Here we have the table, and in the first row,"},{"Start":"00:08.970 ","End":"00:11.054","Text":"we have the number of cars per family,"},{"Start":"00:11.054 ","End":"00:12.810","Text":"1, 2, 3, 4, and 5."},{"Start":"00:12.810 ","End":"00:15.270","Text":"In the second row, we have the frequency."},{"Start":"00:15.270 ","End":"00:19.136","Text":"That means that 65 families own 1 car,"},{"Start":"00:19.136 ","End":"00:20.879","Text":"150 families own 2 cars,"},{"Start":"00:20.879 ","End":"00:22.260","Text":"and so on and so forth."},{"Start":"00:22.260 ","End":"00:24.900","Text":"Now, in descriptive statistics,"},{"Start":"00:24.900 ","End":"00:27.330","Text":"we\u0027re dealing with groups."},{"Start":"00:27.330 ","End":"00:29.730","Text":"We want to analyze groups."},{"Start":"00:29.730 ","End":"00:32.145","Text":"Now, what\u0027s our group here?"},{"Start":"00:32.145 ","End":"00:37.790","Text":"Well, that\u0027ll be the families in the community."},{"Start":"00:37.790 ","End":"00:41.600","Text":"We analyze these families by analyzing the variables."},{"Start":"00:41.600 ","End":"00:45.590","Text":"Now what\u0027s the variable that we\u0027re interested in right here?"},{"Start":"00:45.590 ","End":"00:50.210","Text":"Well, that would be the number of cars per family."},{"Start":"00:50.210 ","End":"00:52.850","Text":"Here, x,"},{"Start":"00:52.850 ","End":"00:54.995","Text":"that will be the number of cars per family,"},{"Start":"00:54.995 ","End":"00:58.115","Text":"and f of x, that\u0027s the frequency."},{"Start":"00:58.115 ","End":"01:00.950","Text":"Now, we\u0027re asked to calculate the standard deviation."},{"Start":"01:00.950 ","End":"01:02.060","Text":"Now in order to do that,"},{"Start":"01:02.060 ","End":"01:06.020","Text":"we need to calculate a whole bunch of stuff beforehand."},{"Start":"01:06.020 ","End":"01:08.090","Text":"Let\u0027s get to work."},{"Start":"01:08.090 ","End":"01:12.665","Text":"Now, the first thing that we want to calculate is n,"},{"Start":"01:12.665 ","End":"01:15.230","Text":"the number of families that we\u0027re talking about."},{"Start":"01:15.230 ","End":"01:20.435","Text":"Now that equals to the sum of the frequencies,"},{"Start":"01:20.435 ","End":"01:29.600","Text":"so we\u0027re looking at 65 plus 150 plus 220 plus 140 plus 55."},{"Start":"01:29.600 ","End":"01:34.135","Text":"Now that equals to 630."},{"Start":"01:34.135 ","End":"01:37.625","Text":"The next thing that we want to calculate is the average."},{"Start":"01:37.625 ","End":"01:43.130","Text":"Well, the average is the sum of x_i times f times the frequency"},{"Start":"01:43.130 ","End":"01:48.745","Text":"divided by n. Now that equals to 1 times the frequency."},{"Start":"01:48.745 ","End":"01:54.915","Text":"That\u0027s 65 plus 2 times 150 plus"},{"Start":"01:54.915 ","End":"02:02.510","Text":"3 times 220 plus 4 times 140 plus 5 times 55."},{"Start":"02:02.510 ","End":"02:09.365","Text":"Now, all that, we have to divide that by n, n is 630."},{"Start":"02:09.365 ","End":"02:14.735","Text":"Now that comes out to 2.952."},{"Start":"02:14.735 ","End":"02:23.170","Text":"So this is the average number of cars that a family has in our specific community."},{"Start":"02:23.420 ","End":"02:26.900","Text":"We\u0027re asked to calculate the standard deviation,"},{"Start":"02:26.900 ","End":"02:28.130","Text":"so we\u0027re almost there."},{"Start":"02:28.130 ","End":"02:35.625","Text":"Now we can\u0027t calculate the standard deviation because we don\u0027t have the variance."},{"Start":"02:35.625 ","End":"02:38.075","Text":"Let\u0027s calculate the variance."},{"Start":"02:38.075 ","End":"02:40.820","Text":"What\u0027s the variance? That\u0027s s squared of x."},{"Start":"02:40.820 ","End":"02:43.190","Text":"Now, the definition of the variance,"},{"Start":"02:43.190 ","End":"02:45.380","Text":"when we have a frequency table,"},{"Start":"02:45.380 ","End":"02:55.250","Text":"that\u0027s the sum of the deviations squared times the frequency divided by n. Now,"},{"Start":"02:55.250 ","End":"02:58.460","Text":"as we said, this is a complicated equation."},{"Start":"02:58.460 ","End":"03:03.830","Text":"We have a working equation and that\u0027s the sum of x_i squared"},{"Start":"03:03.830 ","End":"03:10.270","Text":"times f divided by n minus x-bar squared."},{"Start":"03:10.270 ","End":"03:15.140","Text":"This then would be the equation that we\u0027ll use."},{"Start":"03:15.140 ","End":"03:17.430","Text":"Let\u0027s get to work here."},{"Start":"03:19.220 ","End":"03:23.270","Text":"The sum of x_i squared times f,"},{"Start":"03:23.270 ","End":"03:33.560","Text":"that\u0027ll be equal to 1 squared times the frequency 65 plus 2 squared times 150 plus 3"},{"Start":"03:33.560 ","End":"03:40.010","Text":"squared times 220 plus 4 squared times 140 plus 5"},{"Start":"03:40.010 ","End":"03:48.145","Text":"squared times 55 divided by n. Now n here is 630."},{"Start":"03:48.145 ","End":"03:51.015","Text":"Now, what did we forget?"},{"Start":"03:51.015 ","End":"03:57.860","Text":"Exactly. We want to subtract from this the average squared."},{"Start":"03:57.860 ","End":"04:02.729","Text":"Now the average here, that\u0027s 2.952 squared."},{"Start":"04:02.729 ","End":"04:07.380","Text":"Now all that comes out to 1.222."},{"Start":"04:07.380 ","End":"04:09.560","Text":"What\u0027s the units here?"},{"Start":"04:09.560 ","End":"04:12.845","Text":"The units here are cars squared."},{"Start":"04:12.845 ","End":"04:15.800","Text":"Now, this is not an intuitive unit."},{"Start":"04:15.800 ","End":"04:19.345","Text":"That\u0027s why we have the standard deviation."},{"Start":"04:19.345 ","End":"04:23.160","Text":"The standard deviation, well,"},{"Start":"04:23.160 ","End":"04:27.530","Text":"that equals to the square root of the variance."},{"Start":"04:27.530 ","End":"04:29.750","Text":"Now, in our case,"},{"Start":"04:29.750 ","End":"04:34.310","Text":"that\u0027ll be the square root of 1.222,"},{"Start":"04:34.310 ","End":"04:39.650","Text":"and that equals to 1.106 cars."},{"Start":"04:39.650 ","End":"04:42.860","Text":"Now, here\u0027s something that we can understand."},{"Start":"04:42.860 ","End":"04:44.765","Text":"Now, what does this mean?"},{"Start":"04:44.765 ","End":"04:46.355","Text":"What did we get here?"},{"Start":"04:46.355 ","End":"04:51.180","Text":"Well, we know that on average,"},{"Start":"04:51.180 ","End":"04:54.870","Text":"each family has 2.952 cars."},{"Start":"04:54.870 ","End":"04:56.925","Text":"There\u0027s some families who have less cars."},{"Start":"04:56.925 ","End":"05:00.990","Text":"There\u0027s some families who have more cars."},{"Start":"05:00.990 ","End":"05:07.985","Text":"There\u0027s a variance, there is a disbursement of the number of cars amongst families,"},{"Start":"05:07.985 ","End":"05:11.990","Text":"so what does this 1.106?"},{"Start":"05:11.990 ","End":"05:13.505","Text":"What does this value mean?"},{"Start":"05:13.505 ","End":"05:18.770","Text":"Well, all this means is that on average,"},{"Start":"05:18.770 ","End":"05:28.380","Text":"we have the average deviation of 1.106 cars from this average for each family."},{"Start":"05:28.670 ","End":"05:33.470","Text":"Now, let\u0027s take a look at the other question here."},{"Start":"05:33.470 ","End":"05:36.590","Text":"We\u0027re asked to calculate the range of the data."},{"Start":"05:36.590 ","End":"05:40.070","Text":"Well, what\u0027s the range of our data here?"},{"Start":"05:40.070 ","End":"05:46.555","Text":"Well, that\u0027s defined as x_max minus x_min."},{"Start":"05:46.555 ","End":"05:52.955","Text":"Now, what is the maximum value for x and what\u0027s the minimum value for x?"},{"Start":"05:52.955 ","End":"05:54.965","Text":"Well, as we can see,"},{"Start":"05:54.965 ","End":"05:57.500","Text":"here\u0027s our x values right here."},{"Start":"05:57.500 ","End":"06:04.980","Text":"Now the maximum value for x is 5 and the minimum value for x is 1."},{"Start":"06:05.840 ","End":"06:10.650","Text":"That means that x_max, that\u0027ll be 5,"},{"Start":"06:10.650 ","End":"06:13.215","Text":"minus x_min, that\u0027ll be 1,"},{"Start":"06:13.215 ","End":"06:15.405","Text":"so that equals to 4."},{"Start":"06:15.405 ","End":"06:17.840","Text":"Again, the units are cars,"},{"Start":"06:17.840 ","End":"06:21.480","Text":"so this would be the range."}],"ID":13323},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"5m 42s","ChapterTopicVideoID":12845,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12845.jpeg","UploadDate":"2018-07-30T17:52:25.7830000","DurationForVideoObject":"PT5M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this question, we have a telemarketing company that examined"},{"Start":"00:03.000 ","End":"00:06.090","Text":"the seniority in years of its employees."},{"Start":"00:06.090 ","End":"00:11.175","Text":"Now, the average seniority is 4 years and the standard deviation is 2 years."},{"Start":"00:11.175 ","End":"00:16.170","Text":"In descriptive statistics, we\u0027re analyzing groups of things."},{"Start":"00:16.170 ","End":"00:18.510","Text":"Here, what would be our group?"},{"Start":"00:18.510 ","End":"00:24.180","Text":"Well, that would be the employees in the telemarketing company."},{"Start":"00:24.180 ","End":"00:28.515","Text":"Now, what varies from employee to employee, that\u0027s our variable."},{"Start":"00:28.515 ","End":"00:32.045","Text":"Now, what are we interested here? What\u0027s a variable?"},{"Start":"00:32.045 ","End":"00:36.275","Text":"Well, that would be the number of years of seniority."},{"Start":"00:36.275 ","End":"00:39.290","Text":"Now, we\u0027re given that x bar,"},{"Start":"00:39.290 ","End":"00:42.635","Text":"the average years of seniority that equals to 4."},{"Start":"00:42.635 ","End":"00:45.065","Text":"That\u0027s given to us right here."},{"Start":"00:45.065 ","End":"00:48.469","Text":"What about the standard deviation of the seniority?"},{"Start":"00:48.469 ","End":"00:51.220","Text":"Well, that equals to 2 years."},{"Start":"00:51.220 ","End":"00:53.850","Text":"Now, we\u0027re asked,"},{"Start":"00:53.850 ","End":"00:55.610","Text":"will the average increase,"},{"Start":"00:55.610 ","End":"00:59.810","Text":"decrease, or remain unchanged and will the standard deviation increase,"},{"Start":"00:59.810 ","End":"01:03.050","Text":"decrease, or remain unchanged if 2 new employees with"},{"Start":"01:03.050 ","End":"01:08.700","Text":"4 years of seniority each are added to the probability distribution?"},{"Start":"01:09.020 ","End":"01:12.830","Text":"Let\u0027s first take a look at the average."},{"Start":"01:12.830 ","End":"01:23.210","Text":"Now, if we add any number of employees that have the same seniority as the average,"},{"Start":"01:23.210 ","End":"01:26.330","Text":"well, then what would happen to the overall average?"},{"Start":"01:26.330 ","End":"01:30.710","Text":"Well, there would be no change whatsoever."},{"Start":"01:30.710 ","End":"01:36.560","Text":"Now, that\u0027s true for any number of employees that you add to the distribution or"},{"Start":"01:36.560 ","End":"01:40.070","Text":"basically any observation that you add to"},{"Start":"01:40.070 ","End":"01:44.060","Text":"the distribution if the value of the observation equals to the average,"},{"Start":"01:44.060 ","End":"01:47.075","Text":"then the average will not change whatsoever."},{"Start":"01:47.075 ","End":"01:51.850","Text":"Now, what happens to the standard deviation?"},{"Start":"01:51.850 ","End":"01:56.390","Text":"Well, let\u0027s define the standard deviation again."},{"Start":"01:56.390 ","End":"02:01.700","Text":"Now, that\u0027s the measure of disbursement of the data."},{"Start":"02:01.700 ","End":"02:03.485","Text":"That means that on average,"},{"Start":"02:03.485 ","End":"02:08.035","Text":"the observations deviate from the average by 2 years."},{"Start":"02:08.035 ","End":"02:11.180","Text":"That means that the average distance of"},{"Start":"02:11.180 ","End":"02:14.920","Text":"an employee from the average seniority is 2 years."},{"Start":"02:14.920 ","End":"02:20.015","Text":"Now, if I added 2 employees with 0 deviation,"},{"Start":"02:20.015 ","End":"02:25.830","Text":"that means that the standard deviation decreased."},{"Start":"02:26.240 ","End":"02:32.270","Text":"Now, let\u0027s take a look at this from a mathematical perspective."},{"Start":"02:32.270 ","End":"02:35.375","Text":"Let\u0027s take a look at the equations."},{"Start":"02:35.375 ","End":"02:38.225","Text":"Now, let\u0027s take a look at the variance."},{"Start":"02:38.225 ","End":"02:42.125","Text":"Now, what\u0027s the equation for the variance?"},{"Start":"02:42.125 ","End":"02:48.605","Text":"Well, that\u0027s the sum of the deviation squared divided by"},{"Start":"02:48.605 ","End":"02:53.015","Text":"n. If I now"},{"Start":"02:53.015 ","End":"02:58.910","Text":"took the distribution and I added 2 people in the denominator,"},{"Start":"02:58.910 ","End":"03:01.950","Text":"well, that increased."},{"Start":"03:02.000 ","End":"03:08.765","Text":"But if I added these 2 new people but their deviation was 0,"},{"Start":"03:08.765 ","End":"03:11.660","Text":"that means that this in the numerator, well,"},{"Start":"03:11.660 ","End":"03:20.705","Text":"there was no change because the average years of seniority equal to the average."},{"Start":"03:20.705 ","End":"03:24.655","Text":"This deviation right here would be 0."},{"Start":"03:24.655 ","End":"03:28.325","Text":"Actually, the numerator didn\u0027t change at all."},{"Start":"03:28.325 ","End":"03:31.310","Text":"If I have a ratio where there is"},{"Start":"03:31.310 ","End":"03:34.985","Text":"no change in the numerator but the denominator increased,"},{"Start":"03:34.985 ","End":"03:39.230","Text":"then that means that this ratio decreased."},{"Start":"03:39.230 ","End":"03:41.735","Text":"Now, if the variance decreased,"},{"Start":"03:41.735 ","End":"03:45.260","Text":"then that means that the standard deviation decreased because"},{"Start":"03:45.260 ","End":"03:49.700","Text":"the standard deviation is the square root of the variance."},{"Start":"03:49.700 ","End":"03:51.210","Text":"In this section, we\u0027re asked,"},{"Start":"03:51.210 ","End":"03:52.940","Text":"will the average increase, decrease,"},{"Start":"03:52.940 ","End":"03:57.290","Text":"or remain unchanged, and will the standard deviation increase, decrease,"},{"Start":"03:57.290 ","End":"04:02.510","Text":"or remain unchanged if 2 employees are added to the probability distribution but now,"},{"Start":"04:02.510 ","End":"04:07.270","Text":"1 with 0 years of seniority and 1 with 8 years of seniority."},{"Start":"04:07.270 ","End":"04:09.500","Text":"Let\u0027s take a look at the average."},{"Start":"04:09.500 ","End":"04:15.605","Text":"The average was given to us at 4 years of seniority for the whole distribution."},{"Start":"04:15.605 ","End":"04:18.705","Text":"Now, we\u0027re adding 2 new people,"},{"Start":"04:18.705 ","End":"04:24.280","Text":"but now 1 was 0 years of seniority and the other was 8 years of seniority."},{"Start":"04:24.280 ","End":"04:30.300","Text":"The average seniority between the 2 new guys is equal to 4."},{"Start":"04:30.300 ","End":"04:35.180","Text":"That means that for the total average of"},{"Start":"04:35.180 ","End":"04:39.965","Text":"the population or the distribution after we added the 2 new people,"},{"Start":"04:39.965 ","End":"04:42.020","Text":"well, the average does not change,"},{"Start":"04:42.020 ","End":"04:45.695","Text":"so there\u0027s no change for the average."},{"Start":"04:45.695 ","End":"04:48.379","Text":"Now, what happens to the standard deviation?"},{"Start":"04:48.379 ","End":"04:53.495","Text":"Well, the standard deviation was given to us as 2 years."},{"Start":"04:53.495 ","End":"04:57.770","Text":"Now, if we add 2 new people,"},{"Start":"04:57.770 ","End":"05:00.830","Text":"then what happens to the disbursement?"},{"Start":"05:00.830 ","End":"05:03.330","Text":"Well, let\u0027s take a look."},{"Start":"05:03.370 ","End":"05:08.780","Text":"We had 0 years of seniority minus 4."},{"Start":"05:08.780 ","End":"05:10.745","Text":"That\u0027s a deviation from the average,"},{"Start":"05:10.745 ","End":"05:12.710","Text":"that equals to minus 4."},{"Start":"05:12.710 ","End":"05:15.050","Text":"What about the second guy?"},{"Start":"05:15.050 ","End":"05:18.130","Text":"Well, that\u0027s 8 years minus 4,"},{"Start":"05:18.130 ","End":"05:21.070","Text":"that\u0027s the average, that equals to 4 years."},{"Start":"05:21.070 ","End":"05:30.050","Text":"We see that the deviation here is greater than the deviation of the original population."},{"Start":"05:30.050 ","End":"05:32.810","Text":"By adding these 2 new people,"},{"Start":"05:32.810 ","End":"05:37.490","Text":"we\u0027re actually increasing the deviation from the average,"},{"Start":"05:37.490 ","End":"05:42.690","Text":"so the standard deviation here would increase."}],"ID":13324},{"Watched":false,"Name":"Exercise 4","Duration":"2m 17s","ChapterTopicVideoID":12846,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12846.jpeg","UploadDate":"2018-07-30T17:52:31.1630000","DurationForVideoObject":"PT2M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this question, we\u0027re given a list of 5 observations,"},{"Start":"00:03.150 ","End":"00:07.620","Text":"but the deviation from the average is given only for 4 of them."},{"Start":"00:07.620 ","End":"00:12.240","Text":"We\u0027re asked to calculate the variance of the 5 observations also."},{"Start":"00:12.240 ","End":"00:15.840","Text":"The first thing that we need to do is write down what we know."},{"Start":"00:15.840 ","End":"00:18.435","Text":"How many observations do we have?"},{"Start":"00:18.435 ","End":"00:21.420","Text":"5. What are these numbers right here?"},{"Start":"00:21.420 ","End":"00:23.625","Text":"They\u0027re the deviations from the average."},{"Start":"00:23.625 ","End":"00:26.265","Text":"They\u0027re not the observations themselves."},{"Start":"00:26.265 ","End":"00:29.895","Text":"That means that x_1 minus x-bar,"},{"Start":"00:29.895 ","End":"00:31.875","Text":"that equals to minus 1;"},{"Start":"00:31.875 ","End":"00:33.660","Text":"x_2 minus x-bar,"},{"Start":"00:33.660 ","End":"00:35.280","Text":"that equals to 2;"},{"Start":"00:35.280 ","End":"00:37.410","Text":"x_3 minus x-bar,"},{"Start":"00:37.410 ","End":"00:39.370","Text":"that equals to 3;"},{"Start":"00:39.370 ","End":"00:42.065","Text":"x_4 minus x-bar,"},{"Start":"00:42.065 ","End":"00:43.670","Text":"that equals to 2;"},{"Start":"00:43.670 ","End":"00:45.920","Text":"and x_5 minus x-bar,"},{"Start":"00:45.920 ","End":"00:47.015","Text":"well, we don\u0027t know that."},{"Start":"00:47.015 ","End":"00:50.090","Text":"We\u0027ll just write that down as t. Now,"},{"Start":"00:50.090 ","End":"00:54.270","Text":"we know that the sum of the deviations,"},{"Start":"00:54.270 ","End":"00:58.709","Text":"x_i minus x-bar, that equals to 0."},{"Start":"00:58.709 ","End":"01:02.210","Text":"All we have to do is sum up these guys right here,"},{"Start":"01:02.210 ","End":"01:06.755","Text":"equate them to 0, and find the value for t. Let\u0027s do that."},{"Start":"01:06.755 ","End":"01:08.740","Text":"That\u0027ll be minus 1,"},{"Start":"01:08.740 ","End":"01:11.235","Text":"plus 2, plus 3,"},{"Start":"01:11.235 ","End":"01:16.310","Text":"plus 2, plus t. That has to equal to 0."},{"Start":"01:16.310 ","End":"01:22.930","Text":"That means that t here equals to minus 6."},{"Start":"01:23.300 ","End":"01:26.405","Text":"This guy equals to minus 6."},{"Start":"01:26.405 ","End":"01:34.260","Text":"Now we\u0027re asked to calculate the variance of the 5 observations. What\u0027s the variance?"},{"Start":"01:34.260 ","End":"01:36.660","Text":"That\u0027s s squared sub x,"},{"Start":"01:36.660 ","End":"01:45.530","Text":"and that equals to the sum of the deviation squared divided by n. Well,"},{"Start":"01:45.530 ","End":"01:48.320","Text":"we have the deviations right here."},{"Start":"01:48.320 ","End":"01:50.330","Text":"All we need to do is to square them,"},{"Start":"01:50.330 ","End":"01:51.830","Text":"divide by n,"},{"Start":"01:51.830 ","End":"01:54.995","Text":"and then get our variance. Let\u0027s do that."},{"Start":"01:54.995 ","End":"01:57.830","Text":"That equals to minus 1 squared,"},{"Start":"01:57.830 ","End":"02:00.482","Text":"plus 2 squared,"},{"Start":"02:00.482 ","End":"02:02.774","Text":"plus 3 squared,"},{"Start":"02:02.774 ","End":"02:04.778","Text":"plus 2 squared,"},{"Start":"02:04.778 ","End":"02:10.770","Text":"plus minus 6 squared divided by n. Well, n equals to 5."},{"Start":"02:10.770 ","End":"02:15.110","Text":"That equals to 10.8."},{"Start":"02:15.110 ","End":"02:18.480","Text":"This then would be our variance."}],"ID":13325},{"Watched":false,"Name":"Exercise 5 - Parts a-c","Duration":"8m 16s","ChapterTopicVideoID":12847,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12847.jpeg","UploadDate":"2018-07-30T17:52:58.8670000","DurationForVideoObject":"PT8M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"In this question, the number of rooms in"},{"Start":"00:02.190 ","End":"00:05.115","Text":"each apartment is examined in a certain neighborhood."},{"Start":"00:05.115 ","End":"00:08.310","Text":"There are 200 apartments in the neighborhood and we\u0027re asked,"},{"Start":"00:08.310 ","End":"00:12.030","Text":"what\u0027s the average number of rooms per apartment in the neighborhood?"},{"Start":"00:12.030 ","End":"00:18.630","Text":"In descriptive statistics, we examine groups."},{"Start":"00:18.630 ","End":"00:22.575","Text":"Now here our group, what would that be?"},{"Start":"00:22.575 ","End":"00:28.875","Text":"That would be the 200 apartments in a specific neighborhood."},{"Start":"00:28.875 ","End":"00:33.435","Text":"What\u0027s the variable that we\u0027re interested in?"},{"Start":"00:33.435 ","End":"00:39.255","Text":"Well, that\u0027s the number of rooms per apartment."},{"Start":"00:39.255 ","End":"00:45.060","Text":"Now, this is a quantitative variable and it\u0027s a discrete variable."},{"Start":"00:45.640 ","End":"00:48.455","Text":"How many apartments do we have?"},{"Start":"00:48.455 ","End":"00:53.780","Text":"Well, n here, that was given to us, that\u0027s 200 apartments."},{"Start":"00:53.780 ","End":"00:55.730","Text":"Now as we said,"},{"Start":"00:55.730 ","End":"01:02.630","Text":"this is a table of the data that we got and we\u0027re given here, that\u0027s x,"},{"Start":"01:02.630 ","End":"01:04.273","Text":"that\u0027s the number of rooms,"},{"Start":"01:04.273 ","End":"01:08.225","Text":"but here we\u0027re given the proportion of rooms,"},{"Start":"01:08.225 ","End":"01:11.870","Text":"that\u0027s the proportion of the 200 apartments."},{"Start":"01:11.870 ","End":"01:14.065","Text":"Now, what\u0027s the proportion?"},{"Start":"01:14.065 ","End":"01:21.545","Text":"The proportion is the frequency divided by n. First of all,"},{"Start":"01:21.545 ","End":"01:26.120","Text":"I know that the sum of the proportion, well,"},{"Start":"01:26.120 ","End":"01:28.580","Text":"that has to be equal to 1,"},{"Start":"01:28.580 ","End":"01:32.285","Text":"so the first thing that we want to do is want to complete the table."},{"Start":"01:32.285 ","End":"01:34.165","Text":"What do we have here?"},{"Start":"01:34.165 ","End":"01:37.290","Text":"Well, that\u0027s 0.1 plus 0.2 that\u0027s 0.3,"},{"Start":"01:37.290 ","End":"01:39.135","Text":"plus 0.4, that\u0027s 0.7,"},{"Start":"01:39.135 ","End":"01:42.030","Text":"plus 0.15, that\u0027s 0.85."},{"Start":"01:42.030 ","End":"01:44.800","Text":"In order to complete this to 1,"},{"Start":"01:44.800 ","End":"01:49.810","Text":"then this here has to be equal to 0.15 as well."},{"Start":"01:49.810 ","End":"01:55.255","Text":"We\u0027re asked about the average number of rooms per apartment."},{"Start":"01:55.255 ","End":"02:00.420","Text":"When we have a frequency table, the average,"},{"Start":"02:00.420 ","End":"02:04.100","Text":"that would be equal to the sum of x times"},{"Start":"02:04.100 ","End":"02:08.000","Text":"the frequency divided by n. But we don\u0027t have the frequency,"},{"Start":"02:08.000 ","End":"02:10.495","Text":"we have the proportion."},{"Start":"02:10.495 ","End":"02:14.600","Text":"Let\u0027s just add another column here and that\u0027ll be the frequency."},{"Start":"02:14.600 ","End":"02:18.920","Text":"The frequency will be the proportion times n,"},{"Start":"02:18.920 ","End":"02:23.517","Text":"so that\u0027ll be n times the proportion."},{"Start":"02:23.517 ","End":"02:32.265","Text":"That means that we have 0.1 times 200 and that equals to 20."},{"Start":"02:32.265 ","End":"02:35.925","Text":"Here that\u0027ll be 0.2 times 200,"},{"Start":"02:35.925 ","End":"02:37.865","Text":"that equals to 40."},{"Start":"02:37.865 ","End":"02:41.650","Text":"Here that\u0027ll be 0.4 times 200,"},{"Start":"02:41.650 ","End":"02:43.475","Text":"that equals to 80,"},{"Start":"02:43.475 ","End":"02:47.160","Text":"0.15 times 200,"},{"Start":"02:47.160 ","End":"02:49.095","Text":"that equals to 30,"},{"Start":"02:49.095 ","End":"02:52.620","Text":"and 0.15 times 200,"},{"Start":"02:52.620 ","End":"02:54.585","Text":"that also equals to 30."},{"Start":"02:54.585 ","End":"02:59.100","Text":"In total, I have 200 apartments."},{"Start":"02:59.100 ","End":"03:02.905","Text":"Now we\u0027re ready to calculate the average."},{"Start":"03:02.905 ","End":"03:05.345","Text":"Well, that means that I\u0027m have to sum up"},{"Start":"03:05.345 ","End":"03:09.095","Text":"the multiplication of the observation times the frequency."},{"Start":"03:09.095 ","End":"03:12.960","Text":"That\u0027s 1, that\u0027s the observation, times the frequency."},{"Start":"03:12.960 ","End":"03:14.550","Text":"Well, the frequency here is 20,"},{"Start":"03:14.550 ","End":"03:20.710","Text":"so it\u0027d be 1 times 20 plus 2 times 40,"},{"Start":"03:20.720 ","End":"03:24.630","Text":"plus 3 times 80,"},{"Start":"03:24.630 ","End":"03:28.320","Text":"plus 4 times 30,"},{"Start":"03:28.320 ","End":"03:31.125","Text":"plus 5 times 30,"},{"Start":"03:31.125 ","End":"03:36.045","Text":"and that has to be divided by 200."},{"Start":"03:36.045 ","End":"03:41.865","Text":"Here that equals to 3.05 rooms,"},{"Start":"03:41.865 ","End":"03:44.600","Text":"so that\u0027s the average number of rooms in"},{"Start":"03:44.600 ","End":"03:48.800","Text":"an apartment in a neighborhood with 200 apartments."},{"Start":"03:48.800 ","End":"03:53.390","Text":"In this section, we\u0027re asked to calculate the standard deviation of the number of rooms."},{"Start":"03:53.390 ","End":"04:01.430","Text":"Here we have our frequency table and we\u0027re given that n equals 200,"},{"Start":"04:01.430 ","End":"04:03.170","Text":"that\u0027s the number of apartments,"},{"Start":"04:03.170 ","End":"04:13.750","Text":"and we\u0027ve calculated the average number of rooms per apartment to be 3.05 rooms."},{"Start":"04:13.750 ","End":"04:17.390","Text":"We\u0027re asked to calculate the standard deviation,"},{"Start":"04:17.390 ","End":"04:22.790","Text":"but we can\u0027t do that without first calculating the variance. Let\u0027s do that."},{"Start":"04:22.790 ","End":"04:26.310","Text":"That\u0027ll be the S squared of x,"},{"Start":"04:26.310 ","End":"04:32.510","Text":"and that equals to the sum of the deviations squared times"},{"Start":"04:32.510 ","End":"04:39.660","Text":"the frequency divided by n. We have a working formula,"},{"Start":"04:39.660 ","End":"04:43.460","Text":"and that\u0027s the sum of x squared times"},{"Start":"04:43.460 ","End":"04:48.805","Text":"the frequency divided by n minus the average squared."},{"Start":"04:48.805 ","End":"04:52.520","Text":"This would be the formula that we\u0027ll use to calculate"},{"Start":"04:52.520 ","End":"04:57.050","Text":"the variance and then we\u0027ll calculate the standard deviation."},{"Start":"04:57.050 ","End":"05:00.440","Text":"All we have to do now is do the following."},{"Start":"05:00.440 ","End":"05:02.945","Text":"We have x squared,"},{"Start":"05:02.945 ","End":"05:05.205","Text":"that\u0027s 1 squared,"},{"Start":"05:05.205 ","End":"05:06.800","Text":"times the frequency,"},{"Start":"05:06.800 ","End":"05:08.930","Text":"well the frequency is 20,"},{"Start":"05:08.930 ","End":"05:15.920","Text":"plus 2 squared times 40 plus 3 squared times"},{"Start":"05:15.920 ","End":"05:23.445","Text":"80 plus 4 squared times 30 plus 5 squared times 30,"},{"Start":"05:23.445 ","End":"05:26.915","Text":"and all this has to be divided by n,"},{"Start":"05:26.915 ","End":"05:29.640","Text":"n here is 200."},{"Start":"05:31.880 ","End":"05:35.595","Text":"Let\u0027s not forget the average squared,"},{"Start":"05:35.595 ","End":"05:40.215","Text":"so that\u0027ll be minus 3.05 squared."},{"Start":"05:40.215 ","End":"05:46.455","Text":"All that comes out to 1.3475."},{"Start":"05:46.455 ","End":"05:47.870","Text":"What\u0027s the units here?"},{"Start":"05:47.870 ","End":"05:51.975","Text":"The units are in rooms squared,"},{"Start":"05:51.975 ","End":"05:53.970","Text":"that\u0027s the units of the variance."},{"Start":"05:53.970 ","End":"05:55.530","Text":"This isn\u0027t very intuitive,"},{"Start":"05:55.530 ","End":"05:59.059","Text":"so let\u0027s calculate the standard deviation,"},{"Start":"05:59.059 ","End":"06:01.130","Text":"which is what we\u0027re asked to do."},{"Start":"06:01.130 ","End":"06:03.485","Text":"The standard deviation,"},{"Start":"06:03.485 ","End":"06:08.045","Text":"that equals to the square root of the variance."},{"Start":"06:08.045 ","End":"06:18.800","Text":"That equals to the square root of 1.3475 and that equals to 1.16."},{"Start":"06:18.800 ","End":"06:22.255","Text":"Now the units here are rooms."},{"Start":"06:22.255 ","End":"06:24.915","Text":"This then is a standard deviation."},{"Start":"06:24.915 ","End":"06:26.610","Text":"Let\u0027s explain that."},{"Start":"06:26.610 ","End":"06:32.220","Text":"This value is the value that on average"},{"Start":"06:32.220 ","End":"06:38.600","Text":"each apartment is deviated from the average by 1.16 rooms."},{"Start":"06:38.600 ","End":"06:42.500","Text":"There we have it. This is the standard deviation here."},{"Start":"06:42.500 ","End":"06:45.860","Text":"In this section, we have some owners of"},{"Start":"06:45.860 ","End":"06:48.380","Text":"a 2-room apartment who knocked out a wall"},{"Start":"06:48.380 ","End":"06:51.515","Text":"and converted their apartment to a 1-room apartment."},{"Start":"06:51.515 ","End":"06:53.510","Text":"We\u0027re asked, how would this affect?"},{"Start":"06:53.510 ","End":"06:54.920","Text":"Will it increase, decrease,"},{"Start":"06:54.920 ","End":"06:59.620","Text":"or will it not change the average and the standard deviation?"},{"Start":"06:59.620 ","End":"07:01.880","Text":"Let\u0027s take a look at the average."},{"Start":"07:01.880 ","End":"07:06.050","Text":"Now the average was basically the average number of rooms per"},{"Start":"07:06.050 ","End":"07:11.725","Text":"apartment in the neighborhood. What happened here?"},{"Start":"07:11.725 ","End":"07:17.430","Text":"We had some 2-room apartments and they were converted into a 1-room apartment,"},{"Start":"07:17.430 ","End":"07:23.360","Text":"so that means that the average number of rooms per apartment decreased."},{"Start":"07:23.360 ","End":"07:26.105","Text":"We took some of the observations,"},{"Start":"07:26.105 ","End":"07:32.420","Text":"we decreased the value of the observation so that means that the average also decreased."},{"Start":"07:32.420 ","End":"07:37.740","Text":"Now, what happens to the standard deviation?"},{"Start":"07:37.740 ","End":"07:41.300","Text":"Well, the standard deviation as we said is a measure of disbursement."},{"Start":"07:41.300 ","End":"07:45.935","Text":"It tells me on average how far I am from the average."},{"Start":"07:45.935 ","End":"07:50.655","Text":"Now the average was calculated at 3.05,"},{"Start":"07:50.655 ","End":"07:53.960","Text":"that was the average number of rooms per apartment."},{"Start":"07:53.960 ","End":"07:55.970","Text":"Now what happened here?"},{"Start":"07:55.970 ","End":"08:03.470","Text":"I took a 2-room apartment and I converted it into a 1-room apartment. What did I do?"},{"Start":"08:03.470 ","End":"08:10.300","Text":"I basically increased the distance of this observation from the average,"},{"Start":"08:10.300 ","End":"08:16.530","Text":"so that means that the standard deviation increased."}],"ID":13326},{"Watched":false,"Name":"Exercise 6","Duration":"5m 33s","ChapterTopicVideoID":12848,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12848.jpeg","UploadDate":"2018-07-30T17:53:26.9030000","DurationForVideoObject":"PT5M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this question, we have a table that displays"},{"Start":"00:02.400 ","End":"00:06.435","Text":"the probability distribution of weight in kilograms for a certain group."},{"Start":"00:06.435 ","End":"00:11.085","Text":"We\u0027re asked, what\u0027s the standard deviation of the weight probability distribution?"},{"Start":"00:11.085 ","End":"00:14.445","Text":"In descriptive statistics, we\u0027re looking at groups."},{"Start":"00:14.445 ","End":"00:19.965","Text":"Here, my group would be a group of people."},{"Start":"00:19.965 ","End":"00:22.755","Text":"What\u0027s my variable?"},{"Start":"00:22.755 ","End":"00:25.890","Text":"That\u0027s the thing that varies from person to person."},{"Start":"00:25.890 ","End":"00:28.930","Text":"That\u0027ll be the weight,"},{"Start":"00:29.270 ","End":"00:31.725","Text":"that\u0027ll be in kilograms."},{"Start":"00:31.725 ","End":"00:34.790","Text":"Here we have x,"},{"Start":"00:34.790 ","End":"00:37.235","Text":"that\u0027ll be the weight, and here,"},{"Start":"00:37.235 ","End":"00:40.435","Text":"that\u0027ll be f of x, that\u0027s the frequency."},{"Start":"00:40.435 ","End":"00:46.070","Text":"We see here the weight split up into categories so we here have the frequency."},{"Start":"00:46.070 ","End":"00:50.090","Text":"We know that 10 people weigh between 40 and 45 kilos."},{"Start":"00:50.090 ","End":"00:54.079","Text":"20 people weigh between 45 and 50 kilograms,"},{"Start":"00:54.079 ","End":"00:55.840","Text":"and so on and so forth."},{"Start":"00:55.840 ","End":"00:58.970","Text":"The first thing that I want to do is I want to"},{"Start":"00:58.970 ","End":"01:02.570","Text":"see how many people are we talking about, so n here."},{"Start":"01:02.570 ","End":"01:06.430","Text":"That\u0027s the sum of these guys right here,"},{"Start":"01:06.430 ","End":"01:09.005","Text":"and that comes out to 90."},{"Start":"01:09.005 ","End":"01:14.525","Text":"The next thing that I want to do is I want to calculate the average, that\u0027s x-bar."},{"Start":"01:14.525 ","End":"01:21.545","Text":"That equals to the sum of x times f divided by n. That\u0027ll be what?"},{"Start":"01:21.545 ","End":"01:27.770","Text":"We don\u0027t have a specific value for x right now because we have categories here."},{"Start":"01:27.770 ","End":"01:32.930","Text":"What we need to do is we need to calculate the midpoints of each category."},{"Start":"01:32.930 ","End":"01:37.970","Text":"The midpoint of each category would be like this."},{"Start":"01:37.970 ","End":"01:41.150","Text":"That\u0027ll be 45 plus 40 divided by 2."},{"Start":"01:41.150 ","End":"01:45.980","Text":"That equals here to 42.5;"},{"Start":"01:45.980 ","End":"01:49.670","Text":"here, that\u0027ll be 47.5; here,"},{"Start":"01:49.670 ","End":"01:52.010","Text":"that\u0027ll be 55; here,"},{"Start":"01:52.010 ","End":"01:55.141","Text":"that\u0027ll be 62.5; and here,"},{"Start":"01:55.141 ","End":"01:59.560","Text":"that will be 67.5."},{"Start":"01:59.560 ","End":"02:02.660","Text":"Now I can calculate this,"},{"Start":"02:02.660 ","End":"02:09.380","Text":"and that\u0027ll be 42.5 times 10"},{"Start":"02:09.380 ","End":"02:18.305","Text":"plus 47.5 times 20 plus and so on and so forth for each one of the categories,"},{"Start":"02:18.305 ","End":"02:22.354","Text":"divided by n. Well, n equals 90,"},{"Start":"02:22.354 ","End":"02:28.445","Text":"so that equals to 55 and the units here would be kilograms."},{"Start":"02:28.445 ","End":"02:32.360","Text":"If we take a look at this distribution,"},{"Start":"02:32.360 ","End":"02:37.685","Text":"we could see that we didn\u0027t have to go through all this calculation. Why is that?"},{"Start":"02:37.685 ","End":"02:40.729","Text":"We see here that this distribution is symmetrical."},{"Start":"02:40.729 ","End":"02:45.350","Text":"We have 10 people here and 10 people here,"},{"Start":"02:45.350 ","End":"02:47.390","Text":"20 people here and 20 people here,"},{"Start":"02:47.390 ","End":"02:49.100","Text":"and the 30 in the middle."},{"Start":"02:49.100 ","End":"02:56.990","Text":"That means that the average would be the midpoint of the middle category."},{"Start":"02:56.990 ","End":"03:01.490","Text":"That\u0027s the characteristic of a symmetrical distribution."},{"Start":"03:01.490 ","End":"03:05.285","Text":"Here we have the midpoint here, that\u0027s 55."},{"Start":"03:05.285 ","End":"03:09.000","Text":"Here we\u0027ve calculated to be 55."},{"Start":"03:09.320 ","End":"03:12.290","Text":"Now that we know the averages,"},{"Start":"03:12.290 ","End":"03:15.980","Text":"let\u0027s calculate the standard deviation."},{"Start":"03:15.980 ","End":"03:21.240","Text":"We can\u0027t do that without first calculating the variance."},{"Start":"03:21.290 ","End":"03:23.685","Text":"What\u0027s the variance?"},{"Start":"03:23.685 ","End":"03:26.570","Text":"That\u0027s s squared of x."},{"Start":"03:26.570 ","End":"03:32.629","Text":"That\u0027s defined as the sum of the deviations squared"},{"Start":"03:32.629 ","End":"03:38.735","Text":"times the frequency divided by n. We have a working equation for that."},{"Start":"03:38.735 ","End":"03:45.950","Text":"That\u0027ll be the sum of x squared times f divided by n minus x-bar squared."},{"Start":"03:45.950 ","End":"03:52.070","Text":"Let\u0027s use this working equation right here to calculate this guy right here."},{"Start":"03:52.070 ","End":"03:54.980","Text":"That\u0027ll be x squared."},{"Start":"03:54.980 ","End":"03:56.510","Text":"We\u0027ll use the midpoints again,"},{"Start":"03:56.510 ","End":"04:03.035","Text":"so that\u0027ll be 42.5 squared times the frequency, that\u0027s 10,"},{"Start":"04:03.035 ","End":"04:09.515","Text":"plus 47.5 squared times 20,"},{"Start":"04:09.515 ","End":"04:11.435","Text":"and so on and so forth,"},{"Start":"04:11.435 ","End":"04:15.680","Text":"divided by n, n is 90."},{"Start":"04:15.680 ","End":"04:20.640","Text":"Let\u0027s not forget to take away x-bar squared."},{"Start":"04:20.640 ","End":"04:21.780","Text":"What\u0027s x-bar?"},{"Start":"04:21.780 ","End":"04:26.370","Text":"x-bar is 55, so 55 squared."},{"Start":"04:26.370 ","End":"04:31.610","Text":"That equals to 59.72."},{"Start":"04:31.610 ","End":"04:34.819","Text":"The units here are in kilograms squared,"},{"Start":"04:34.819 ","End":"04:36.550","Text":"which isn\u0027t that intuitive."},{"Start":"04:36.550 ","End":"04:41.160","Text":"What we need to do is we need to calculate the standard deviation."},{"Start":"04:41.780 ","End":"04:45.185","Text":"The standard deviation, that\u0027s s,"},{"Start":"04:45.185 ","End":"04:48.275","Text":"that equals to the square root of the variance,"},{"Start":"04:48.275 ","End":"04:53.600","Text":"that equals to the square root of 59.72,"},{"Start":"04:53.600 ","End":"04:59.575","Text":"and that equals to 7.73 kilograms."},{"Start":"04:59.575 ","End":"05:04.310","Text":"This then is what we were asked to do to find the standard deviation."},{"Start":"05:04.310 ","End":"05:06.035","Text":"What is that?"},{"Start":"05:06.035 ","End":"05:14.160","Text":"We know that the standard deviation is a measure of dispersion and that\u0027s"},{"Start":"05:14.160 ","End":"05:23.195","Text":"the average distance from the average of the observations for each observation."},{"Start":"05:23.195 ","End":"05:25.430","Text":"That means that on average,"},{"Start":"05:25.430 ","End":"05:34.860","Text":"each observation is 7.73 kilograms away from the average of 55 kilos."}],"ID":13327},{"Watched":false,"Name":"Exercise 7 - Parts a-c","Duration":"8m 1s","ChapterTopicVideoID":12849,"CourseChapterTopicPlaylistID":265042,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12849.jpeg","UploadDate":"2018-07-30T17:54:08.1830000","DurationForVideoObject":"PT8M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this question, we\u0027re given a diagram that displays"},{"Start":"00:02.940 ","End":"00:06.410","Text":"the probability distribution of marks on an intelligence test."},{"Start":"00:06.410 ","End":"00:08.303","Text":"We\u0027re asked what is the average,"},{"Start":"00:08.303 ","End":"00:11.340","Text":"and median if the probability distribution?"},{"Start":"00:11.340 ","End":"00:16.185","Text":"In descriptive statistics, we\u0027re looking to analyze groups."},{"Start":"00:16.185 ","End":"00:20.175","Text":"Now in our case, what would be my group here?"},{"Start":"00:20.175 ","End":"00:28.680","Text":"Well, that\u0027s the students taking the IQ exam."},{"Start":"00:28.680 ","End":"00:31.395","Text":"What\u0027s my variable?"},{"Start":"00:31.395 ","End":"00:35.750","Text":"Well, the variable is the thing that varies from student to student,"},{"Start":"00:35.750 ","End":"00:38.740","Text":"and that\u0027s the marks."},{"Start":"00:38.740 ","End":"00:41.570","Text":"We\u0027re looking for the average,"},{"Start":"00:41.570 ","End":"00:44.630","Text":"and median of this probability distribution."},{"Start":"00:44.630 ","End":"00:47.570","Text":"Now, if we take a look at this distribution,"},{"Start":"00:47.570 ","End":"00:50.945","Text":"we see that this is symmetrical."},{"Start":"00:50.945 ","End":"00:54.695","Text":"Now, in a symmetrical distribution,"},{"Start":"00:54.695 ","End":"00:58.685","Text":"the average is equal to the median here."},{"Start":"00:58.685 ","End":"01:01.910","Text":"Now, let\u0027s take a look at what\u0027s happening here."},{"Start":"01:01.910 ","End":"01:07.040","Text":"Well, here, that\u0027s exactly the middle of the distribution right here,"},{"Start":"01:07.040 ","End":"01:09.120","Text":"that equals to a 100."},{"Start":"01:09.120 ","End":"01:14.210","Text":"What we have here is we have the average that equals to the median,"},{"Start":"01:14.210 ","End":"01:17.480","Text":"and that equals to a mark of a 100."},{"Start":"01:17.480 ","End":"01:21.875","Text":"In this section, we\u0027re asked to calculate the standard deviation of the marks."},{"Start":"01:21.875 ","End":"01:24.125","Text":"So how do we do that?"},{"Start":"01:24.125 ","End":"01:26.359","Text":"Well, we have a histogram here,"},{"Start":"01:26.359 ","End":"01:32.030","Text":"and we want to convert that to a frequency table by categories."},{"Start":"01:32.030 ","End":"01:37.639","Text":"Let\u0027s do that, and only then can we calculate the standard deviation."},{"Start":"01:37.639 ","End":"01:39.470","Text":"Let\u0027s see how we do this."},{"Start":"01:39.470 ","End":"01:43.135","Text":"That\u0027ll be our x, and that\u0027ll be our frequency."},{"Start":"01:43.135 ","End":"01:45.330","Text":"Now, what are our categories?"},{"Start":"01:45.330 ","End":"01:52.140","Text":"Well, the categories is 70-80, 80-90,"},{"Start":"01:52.140 ","End":"01:55.480","Text":"90-100,"},{"Start":"01:55.730 ","End":"02:07.530","Text":"100-110,110-120, and 120-130."},{"Start":"02:07.530 ","End":"02:09.905","Text":"Now, what about our frequency?"},{"Start":"02:09.905 ","End":"02:11.434","Text":"Well, this is a histogram."},{"Start":"02:11.434 ","End":"02:13.340","Text":"This doesn\u0027t show me frequency."},{"Start":"02:13.340 ","End":"02:20.970","Text":"But, I do know we\u0027re given that 1 square here that equals to 12 students."},{"Start":"02:20.970 ","End":"02:23.360","Text":"How many squares do I have in this category?"},{"Start":"02:23.360 ","End":"02:25.230","Text":"We\u0027ll have 2 squares."},{"Start":"02:25.230 ","End":"02:28.020","Text":"Well, 2 times 12 that\u0027s 24."},{"Start":"02:28.020 ","End":"02:29.970","Text":"How many squares do I have here?"},{"Start":"02:29.970 ","End":"02:32.355","Text":"Well, that\u0027s 1, 2, 3, 4."},{"Start":"02:32.355 ","End":"02:35.835","Text":"So 4 times 12, that\u0027s 48."},{"Start":"02:35.835 ","End":"02:38.070","Text":"How many squares do I have here?"},{"Start":"02:38.070 ","End":"02:39.360","Text":"Well that\u0027s 1, 2, 3,"},{"Start":"02:39.360 ","End":"02:40.500","Text":"4, 5, 6,"},{"Start":"02:40.500 ","End":"02:44.360","Text":"7, 8, 8 times 12, that\u0027s 96."},{"Start":"02:44.360 ","End":"02:46.190","Text":"Now because it\u0027s symmetrical,"},{"Start":"02:46.190 ","End":"02:47.945","Text":"this distribution is symmetrical,"},{"Start":"02:47.945 ","End":"02:49.765","Text":"then this would be 96,"},{"Start":"02:49.765 ","End":"02:52.580","Text":"48, and 24."},{"Start":"02:52.580 ","End":"02:55.715","Text":"Now, what do I want to calculate now?"},{"Start":"02:55.715 ","End":"02:58.580","Text":"I want to know how many students are there."},{"Start":"02:58.580 ","End":"03:06.055","Text":"Well, all I have to do is just sum up the frequencies and that comes up to 336."},{"Start":"03:06.055 ","End":"03:08.925","Text":"Now I know what my average is."},{"Start":"03:08.925 ","End":"03:12.170","Text":"I\u0027ve calculated then the last section. That\u0027s a 100."},{"Start":"03:12.170 ","End":"03:15.470","Text":"Now I can go ahead,"},{"Start":"03:15.470 ","End":"03:17.585","Text":"and calculate the standard deviation."},{"Start":"03:17.585 ","End":"03:19.070","Text":"But in order to do that,"},{"Start":"03:19.070 ","End":"03:23.480","Text":"I need to find out what my variance is. Let\u0027s do that."},{"Start":"03:23.480 ","End":"03:26.300","Text":"The variance s squared of x,"},{"Start":"03:26.300 ","End":"03:30.470","Text":"well that equals to the sum of the deviations"},{"Start":"03:30.470 ","End":"03:37.160","Text":"xi minus x bar squared times the frequency divided by n. Now,"},{"Start":"03:37.160 ","End":"03:39.635","Text":"we\u0027ll use our working formula,"},{"Start":"03:39.635 ","End":"03:49.520","Text":"and that equals to the sum of xi squared times f divided by n minus x bar squared."},{"Start":"03:49.520 ","End":"03:51.830","Text":"Now, in order to calculate this,"},{"Start":"03:51.830 ","End":"03:55.520","Text":"I need to know what the midpoints of these categories are."},{"Start":"03:55.520 ","End":"03:57.965","Text":"Let\u0027s just calculate that here."},{"Start":"03:57.965 ","End":"04:00.740","Text":"There\u0027ll be the midpoint."},{"Start":"04:00.740 ","End":"04:05.810","Text":"Now, the midpoint between 70 and 80 would be 75."},{"Start":"04:05.810 ","End":"04:08.510","Text":"In here, that\u0027ll be 85."},{"Start":"04:08.510 ","End":"04:12.650","Text":"That\u0027ll be 95. That\u0027ll be a 105,"},{"Start":"04:12.650 ","End":"04:16.540","Text":"115, and 125."},{"Start":"04:16.540 ","End":"04:22.080","Text":"Let\u0027s just now calculate this expression right here."},{"Start":"04:26.030 ","End":"04:28.965","Text":"That would be equal to,"},{"Start":"04:28.965 ","End":"04:30.830","Text":"now, xi squared."},{"Start":"04:30.830 ","End":"04:32.390","Text":"Well, that\u0027s the midpoint square,"},{"Start":"04:32.390 ","End":"04:36.170","Text":"so that\u0027ll be 75 squared times the frequency,"},{"Start":"04:36.170 ","End":"04:43.205","Text":"times 24 plus 85 squared times 48,"},{"Start":"04:43.205 ","End":"04:49.475","Text":"plus 95 squared times 96 plus,"},{"Start":"04:49.475 ","End":"04:57.720","Text":"and so on and so forth divided by n. Well n is 336."},{"Start":"04:57.720 ","End":"05:03.080","Text":"Now, from this we have to subtract the average squared,"},{"Start":"05:03.080 ","End":"05:12.710","Text":"100 squared, and that comes out to 167.86."},{"Start":"05:12.710 ","End":"05:17.030","Text":"Now, the units here are in marks squared."},{"Start":"05:17.030 ","End":"05:21.590","Text":"This isn\u0027t an intuitive measure."},{"Start":"05:21.590 ","End":"05:24.469","Text":"Let\u0027s take the standard deviation."},{"Start":"05:24.469 ","End":"05:27.574","Text":"Well, the standard deviation,"},{"Start":"05:27.574 ","End":"05:31.955","Text":"that equals to the square root of the variance,"},{"Start":"05:31.955 ","End":"05:38.450","Text":"that equals to the square root of 167.86,"},{"Start":"05:38.450 ","End":"05:46.145","Text":"and that equals to 12.96 marks."},{"Start":"05:46.145 ","End":"05:48.800","Text":"Now, what does that mean?"},{"Start":"05:48.800 ","End":"05:51.950","Text":"Well, we have different students,"},{"Start":"05:51.950 ","End":"05:54.620","Text":"and each student got a different mark."},{"Start":"05:54.620 ","End":"05:58.220","Text":"Now, that means this value right here,"},{"Start":"05:58.220 ","End":"06:01.595","Text":"means that on average,"},{"Start":"06:01.595 ","End":"06:09.920","Text":"each student deviated from the average of the marks by 12.96 marks."},{"Start":"06:09.920 ","End":"06:13.610","Text":"In this section, we\u0027re given that 20 observations must"},{"Start":"06:13.610 ","End":"06:17.068","Text":"be added to both the 90-100 category,"},{"Start":"06:17.068 ","End":"06:19.985","Text":"and to the 100-110 category."},{"Start":"06:19.985 ","End":"06:24.550","Text":"We\u0027re asked how would this affect the average and the standard deviation."},{"Start":"06:24.550 ","End":"06:30.500","Text":"Basically what we\u0027re doing is we\u0027re taking these 2 categories right here,"},{"Start":"06:30.500 ","End":"06:37.775","Text":"and we\u0027re adding 20 more people symmetrically to the center of the distribution."},{"Start":"06:37.775 ","End":"06:44.090","Text":"Now, what happens to the average when we do that?"},{"Start":"06:44.090 ","End":"06:48.440","Text":"Well, because this is a symmetrical distribution,"},{"Start":"06:48.440 ","End":"06:55.545","Text":"there will be no change to the average."},{"Start":"06:55.545 ","End":"06:58.665","Text":"Because it\u0027s symmetrical,"},{"Start":"06:58.665 ","End":"07:01.093","Text":"this distribution is symmetrical,"},{"Start":"07:01.093 ","End":"07:06.140","Text":"and we\u0027re adding observations to the 2 central categories."},{"Start":"07:06.140 ","End":"07:09.125","Text":"Well, nothing has changed, nothing has moved."},{"Start":"07:09.125 ","End":"07:13.960","Text":"The middle of the distribution is still at a 100."},{"Start":"07:13.960 ","End":"07:20.495","Text":"That means that the average is equal to a 100 and the median is also equal to a 100."},{"Start":"07:20.495 ","End":"07:24.950","Text":"Now, what happens to the standard deviation?"},{"Start":"07:24.950 ","End":"07:32.990","Text":"Well, when we add more observations to the center of the distribution,"},{"Start":"07:32.990 ","End":"07:38.255","Text":"then the standard deviation decreases."},{"Start":"07:38.255 ","End":"07:39.995","Text":"Now, why is that?"},{"Start":"07:39.995 ","End":"07:44.480","Text":"Well, that means that the distribution becomes more centralized,"},{"Start":"07:44.480 ","End":"07:46.925","Text":"more observations are in the middle,"},{"Start":"07:46.925 ","End":"07:49.850","Text":"so the standard deviation becomes smaller."},{"Start":"07:49.850 ","End":"07:53.135","Text":"There\u0027s less disbursement of the data."},{"Start":"07:53.135 ","End":"07:57.608","Text":"Everything becomes more central around the average,"},{"Start":"07:57.608 ","End":"08:02.580","Text":"and as such, the standard deviation decreases."}],"ID":13328}],"Thumbnail":null,"ID":265042},{"Name":"Measures of Dispersion - Interquartile Range","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"3m ","ChapterTopicVideoID":12850,"CourseChapterTopicPlaylistID":265043,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12850.jpeg","UploadDate":"2018-07-30T17:56:45.0730000","DurationForVideoObject":"PT3M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this chapter, I\u0027ll be talking about Descriptive Statistics,"},{"Start":"00:02.820 ","End":"00:04.770","Text":"specifically Measures of Dispersion."},{"Start":"00:04.770 ","End":"00:07.230","Text":"I\u0027ll be talking about the Inter-quartile Range."},{"Start":"00:07.230 ","End":"00:14.190","Text":"Now, this is another measure that belongs to the Measures of Dispersion,"},{"Start":"00:14.190 ","End":"00:15.975","Text":"just like the distribution range,"},{"Start":"00:15.975 ","End":"00:18.465","Text":"the variance and the standard deviation."},{"Start":"00:18.465 ","End":"00:21.840","Text":"Now, let\u0027s say I have a distribution here."},{"Start":"00:21.840 ","End":"00:23.850","Text":"This would be f of x."},{"Start":"00:23.850 ","End":"00:26.610","Text":"That\u0027s the frequency, and here I\u0027ll have x."},{"Start":"00:26.610 ","End":"00:29.820","Text":"Now, these 2 points right here,"},{"Start":"00:29.820 ","End":"00:31.755","Text":"they\u0027ll be X_max,"},{"Start":"00:31.755 ","End":"00:34.560","Text":"the maximum value in the distribution,"},{"Start":"00:34.560 ","End":"00:36.780","Text":"and here they\u0027ll be X_min,"},{"Start":"00:36.780 ","End":"00:39.515","Text":"that will be the minimum value in my distribution."},{"Start":"00:39.515 ","End":"00:41.570","Text":"Now, what I want to do is,"},{"Start":"00:41.570 ","End":"00:49.325","Text":"I want to divide the area under this curve into 4 equal parts."},{"Start":"00:49.325 ","End":"00:54.195","Text":"That means that I\u0027ll have this thing right here."},{"Start":"00:54.195 ","End":"00:59.745","Text":"Where here, I\u0027ll have 1/4 of my observations,"},{"Start":"00:59.745 ","End":"01:02.745","Text":"here I\u0027ll have 1/4 of my observations,"},{"Start":"01:02.745 ","End":"01:05.670","Text":"here they\u0027ll also be 1/4 of my observations,"},{"Start":"01:05.670 ","End":"01:08.835","Text":"and here that\u0027s also 1/4 my observations."},{"Start":"01:08.835 ","End":"01:12.435","Text":"This x value, I\u0027ll call Q_1."},{"Start":"01:12.435 ","End":"01:14.340","Text":"Here, I\u0027ll say Q_2,"},{"Start":"01:14.340 ","End":"01:16.335","Text":"and that will be Q_3."},{"Start":"01:16.335 ","End":"01:19.455","Text":"Now, let\u0027s take a look at Q_1."},{"Start":"01:19.455 ","End":"01:21.705","Text":"What is Q_1?"},{"Start":"01:21.705 ","End":"01:24.005","Text":"Q_1 is the x value,"},{"Start":"01:24.005 ","End":"01:27.950","Text":"where 1/4 of the observations fall below this value,"},{"Start":"01:27.950 ","End":"01:32.060","Text":"and 3/4 of the observations fall above this value."},{"Start":"01:32.060 ","End":"01:38.885","Text":"Now, this is also called X_0.25 or the 25th percentile."},{"Start":"01:38.885 ","End":"01:41.149","Text":"Again, why 25th percentile?"},{"Start":"01:41.149 ","End":"01:47.275","Text":"Because 25 percent of the observations fall below this value right here."},{"Start":"01:47.275 ","End":"01:49.515","Text":"What about Q_3?"},{"Start":"01:49.515 ","End":"01:53.095","Text":"Q_3 is the x value,"},{"Start":"01:53.095 ","End":"01:58.880","Text":"where 75 percent of the observations fall below this value,"},{"Start":"01:58.880 ","End":"02:02.890","Text":"and 25 percent fall above this value."},{"Start":"02:02.890 ","End":"02:10.085","Text":"Now this is called also X_0.75 or the 75th percentile."},{"Start":"02:10.085 ","End":"02:14.170","Text":"Now, another x value that\u0027s important,"},{"Start":"02:14.170 ","End":"02:17.660","Text":"but not relevant to our discussion here is Q_2."},{"Start":"02:17.660 ","End":"02:23.820","Text":"Well, Q_2 is the x value where 1/2 of the population falls below this value,"},{"Start":"02:23.820 ","End":"02:26.385","Text":"and 1/2 falls above this value."},{"Start":"02:26.385 ","End":"02:28.850","Text":"We know that to be the median."},{"Start":"02:28.850 ","End":"02:31.685","Text":"Now the Inter-quartile Range,"},{"Start":"02:31.685 ","End":"02:34.460","Text":"or the IQR,"},{"Start":"02:34.460 ","End":"02:40.435","Text":"that equals to Q_3 minus Q_1."},{"Start":"02:40.435 ","End":"02:46.600","Text":"We\u0027re looking at the middle half of the population here."},{"Start":"02:46.600 ","End":"02:49.680","Text":"That\u0027s this area right here."},{"Start":"02:49.680 ","End":"02:55.470","Text":"Now, this is 1 of the Measures of Dispersion."},{"Start":"02:55.470 ","End":"03:01.410","Text":"Let\u0027s see how we can calculate the Inter-quartile Range."}],"ID":13367},{"Watched":false,"Name":"Tutorial 2","Duration":"5m 46s","ChapterTopicVideoID":12851,"CourseChapterTopicPlaylistID":265043,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12851.jpeg","UploadDate":"2018-07-30T17:57:05.9400000","DurationForVideoObject":"PT5M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Here we have a frequency table by categories,"},{"Start":"00:03.270 ","End":"00:08.580","Text":"that describes the frequency of employees by their seniority in a company."},{"Start":"00:08.580 ","End":"00:12.390","Text":"For example, I have 56 employees who"},{"Start":"00:12.390 ","End":"00:16.140","Text":"have been with the company for between half-year and 4.5 years."},{"Start":"00:16.140 ","End":"00:17.460","Text":"That\u0027s the first category."},{"Start":"00:17.460 ","End":"00:19.320","Text":"Then the second category,"},{"Start":"00:19.320 ","End":"00:23.280","Text":"I have 50 employees who\u0027ve been with the company for between"},{"Start":"00:23.280 ","End":"00:27.900","Text":"4.5 and 9.5 years and so on and so forth."},{"Start":"00:27.900 ","End":"00:31.440","Text":"I\u0027ve also included the width of the category."},{"Start":"00:31.440 ","End":"00:33.780","Text":"For example, in the first category,"},{"Start":"00:33.780 ","End":"00:37.850","Text":"they\u0027ll be 4.5 minus a half a year, and that\u0027s 4 years."},{"Start":"00:37.850 ","End":"00:41.075","Text":"The second category they\u0027ll be 9.5 minus 4.5,"},{"Start":"00:41.075 ","End":"00:43.465","Text":"so that\u0027s 5 years."},{"Start":"00:43.465 ","End":"00:50.660","Text":"The first thing that I want to do is I want to know where my Q1 and Q3 are,"},{"Start":"00:50.660 ","End":"00:54.830","Text":"in which categories are these quartiles."},{"Start":"00:54.830 ","End":"00:56.810","Text":"How do I calculate that?"},{"Start":"00:56.810 ","End":"01:01.775","Text":"Well, to calculate the location of the bottom quartile Q1,"},{"Start":"01:01.775 ","End":"01:06.500","Text":"well, let\u0027s take a look at how many employees I have here."},{"Start":"01:06.500 ","End":"01:10.580","Text":"Well, I have here 200 employees."},{"Start":"01:10.580 ","End":"01:16.715","Text":"The Q1 is the x value"},{"Start":"01:16.715 ","End":"01:21.425","Text":"where a quarter of the employees are below that value."},{"Start":"01:21.425 ","End":"01:27.320","Text":"I have to take 200 and I have to divide that by 4 and that equals to 50."},{"Start":"01:27.320 ","End":"01:31.805","Text":"I\u0027m looking right for the first 50 people."},{"Start":"01:31.805 ","End":"01:35.915","Text":"Now, let\u0027s take a look right here at this column."},{"Start":"01:35.915 ","End":"01:41.525","Text":"This is the cumulative frequency of the employees here."},{"Start":"01:41.525 ","End":"01:42.920","Text":"In the first category,"},{"Start":"01:42.920 ","End":"01:44.675","Text":"I have 56 employees."},{"Start":"01:44.675 ","End":"01:45.930","Text":"Here, well,"},{"Start":"01:45.930 ","End":"01:49.530","Text":"that\u0027s the number of employees in the first and second category."},{"Start":"01:49.530 ","End":"01:52.170","Text":"Here, that\u0027ll be the number of employees in the first,"},{"Start":"01:52.170 ","End":"01:55.025","Text":"in second, and third category and so on and so forth."},{"Start":"01:55.025 ","End":"01:58.535","Text":"For Q1, I\u0027m looking for the 50th employee."},{"Start":"01:58.535 ","End":"02:00.895","Text":"Well, that\u0027s right here."},{"Start":"02:00.895 ","End":"02:07.170","Text":"That means that this would be the category for Q1. What about Q3?"},{"Start":"02:07.170 ","End":"02:11.340","Text":"Well, Q3 is the x-value where 3/4"},{"Start":"02:11.340 ","End":"02:16.310","Text":"of the employees are below that value and are quarter above that value."},{"Start":"02:16.310 ","End":"02:19.720","Text":"Here, that will be 3n divided by 4,"},{"Start":"02:19.720 ","End":"02:22.955","Text":"that will be 3 times 200 divided by 4,"},{"Start":"02:22.955 ","End":"02:24.755","Text":"and that equals to 150."},{"Start":"02:24.755 ","End":"02:28.535","Text":"Here, I\u0027m looking for the 150th person."},{"Start":"02:28.535 ","End":"02:33.800","Text":"Well, that\u0027s right here in this category right here."},{"Start":"02:33.800 ","End":"02:41.269","Text":"I know that Q3 would be in this category and Q1 would be in this category."},{"Start":"02:41.269 ","End":"02:43.805","Text":"Having found these categories,"},{"Start":"02:43.805 ","End":"02:50.000","Text":"now I go to calculate what the value of Q1 and Q3 are. Let\u0027s do that."},{"Start":"02:50.000 ","End":"02:52.505","Text":"Here, I have the formula for"},{"Start":"02:52.505 ","End":"02:56.270","Text":"Q1 and this looks like a daunting expression, but really isn\u0027t."},{"Start":"02:56.270 ","End":"02:58.760","Text":"If we recall from the last chapter,"},{"Start":"02:58.760 ","End":"03:05.145","Text":"this is very similar to calculating the median. Let\u0027s get to work."},{"Start":"03:05.145 ","End":"03:09.960","Text":"Well, for Q1, what\u0027s L_0?"},{"Start":"03:09.960 ","End":"03:15.270","Text":"That\u0027s the lower bound of the category where we found Q1."},{"Start":"03:15.270 ","End":"03:18.660","Text":"Well, we found Q1 in this category right here."},{"Start":"03:18.660 ","End":"03:21.740","Text":"What\u0027s the lower bound? That\u0027s 0.5."},{"Start":"03:21.740 ","End":"03:27.890","Text":"That\u0027s this value right here plus now n divided by 4."},{"Start":"03:27.890 ","End":"03:32.060","Text":"Well, we\u0027ve calculated that, that\u0027s 50 minus."},{"Start":"03:32.060 ","End":"03:33.740","Text":"Now, big F,"},{"Start":"03:33.740 ","End":"03:37.250","Text":"that\u0027s a cumulative frequency of the preceding category."},{"Start":"03:37.250 ","End":"03:39.439","Text":"Well, since this is a first category,"},{"Start":"03:39.439 ","End":"03:41.570","Text":"the preceding category is 0,"},{"Start":"03:41.570 ","End":"03:43.985","Text":"we don\u0027t have a preceding category."},{"Start":"03:43.985 ","End":"03:49.100","Text":"Divided by the frequency of the employees in this category."},{"Start":"03:49.100 ","End":"03:53.080","Text":"Well, that\u0027s 56 times,"},{"Start":"03:53.080 ","End":"03:55.320","Text":"well, L_1 minus L_0."},{"Start":"03:55.320 ","End":"03:59.225","Text":"Well, that\u0027s the width of the category."},{"Start":"03:59.225 ","End":"04:03.515","Text":"That\u0027ll be 4.5 minus a half that\u0027s 4, that\u0027s right here."},{"Start":"04:03.515 ","End":"04:09.470","Text":"Now, that equals to 4.07."},{"Start":"04:09.470 ","End":"04:13.865","Text":"Now, similarly, we\u0027ll do the same calculation for Q3."},{"Start":"04:13.865 ","End":"04:17.135","Text":"Now, this is the category right here for Q3."},{"Start":"04:17.135 ","End":"04:18.635","Text":"Let\u0027s get to work here."},{"Start":"04:18.635 ","End":"04:23.945","Text":"The lower bound of this category is 9.5, that\u0027s right here."},{"Start":"04:23.945 ","End":"04:31.505","Text":"That\u0027ll be 9.5 plus now 3n divided by 4 we\u0027ve calculated that that\u0027s 150."},{"Start":"04:31.505 ","End":"04:34.670","Text":"That\u0027s right here, minus, well,"},{"Start":"04:34.670 ","End":"04:39.605","Text":"what\u0027s the cumulative frequency of the previous category?"},{"Start":"04:39.605 ","End":"04:42.140","Text":"Well, this is the category for Q3,"},{"Start":"04:42.140 ","End":"04:44.765","Text":"the previous category is right here."},{"Start":"04:44.765 ","End":"04:49.490","Text":"This is 106 divided by,"},{"Start":"04:49.490 ","End":"04:53.270","Text":"what\u0027s the frequency of the category where we found Q3?"},{"Start":"04:53.270 ","End":"04:59.270","Text":"Well, that\u0027s 48 times the width of the category."},{"Start":"04:59.270 ","End":"05:03.250","Text":"Well, 11.5 minus 9.5 that\u0027s 2,"},{"Start":"05:03.250 ","End":"05:05.195","Text":"that\u0027s this thing right here."},{"Start":"05:05.195 ","End":"05:09.480","Text":"That equals to 11.33."},{"Start":"05:09.480 ","End":"05:15.420","Text":"Excellent. Now, we found the values for Q1 and Q3,"},{"Start":"05:15.420 ","End":"05:19.865","Text":"so we\u0027re ready to calculate the inter-quartile range."},{"Start":"05:19.865 ","End":"05:26.760","Text":"The inter-quartile range would be the difference between Q3 and Q1."},{"Start":"05:26.760 ","End":"05:29.116","Text":"We have here what\u0027s Q3?"},{"Start":"05:29.116 ","End":"05:30.490","Text":"Well, that\u0027s right here,"},{"Start":"05:30.490 ","End":"05:35.280","Text":"that\u0027s 11.33 minus Q1."},{"Start":"05:35.280 ","End":"05:41.850","Text":"Well, that\u0027s 4.07 so that equals to 7.26."},{"Start":"05:41.850 ","End":"05:47.790","Text":"Here we have it, that\u0027s the value of the inter-quartile range."}],"ID":13368},{"Watched":false,"Name":"Exercise 1","Duration":"5m 48s","ChapterTopicVideoID":12852,"CourseChapterTopicPlaylistID":265043,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12852.jpeg","UploadDate":"2018-07-30T17:57:35.0300000","DurationForVideoObject":"PT5M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this question, we\u0027re given a table that displays"},{"Start":"00:02.280 ","End":"00:06.270","Text":"the probability distribution of weight in kilograms for a specific group,"},{"Start":"00:06.270 ","End":"00:08.760","Text":"and we\u0027re asked to find the inter-quartile range."},{"Start":"00:08.760 ","End":"00:10.095","Text":"Now, what do we have here?"},{"Start":"00:10.095 ","End":"00:14.760","Text":"First of all, we have our x variable and that\u0027s weight."},{"Start":"00:14.760 ","End":"00:16.740","Text":"Here we have the frequency,"},{"Start":"00:16.740 ","End":"00:18.410","Text":"that\u0027s f of x."},{"Start":"00:18.410 ","End":"00:23.160","Text":"So we have 10 people who weigh between 40 and 45 kilograms,"},{"Start":"00:23.160 ","End":"00:27.510","Text":"20 people weigh between 45 and 50 kilograms and so on and so forth."},{"Start":"00:27.510 ","End":"00:31.790","Text":"Now, how many people do we have in total?"},{"Start":"00:31.790 ","End":"00:34.970","Text":"N here that equals to 90."},{"Start":"00:34.970 ","End":"00:38.900","Text":"So the first thing that I need to do in order to find"},{"Start":"00:38.900 ","End":"00:45.110","Text":"the inter-quartile range is to find the categories where the quartiles are located."},{"Start":"00:45.110 ","End":"00:51.034","Text":"So let\u0027s find the location or the category where Q_1 is located."},{"Start":"00:51.034 ","End":"00:56.990","Text":"Well, that means that I have to take n and I have to divide that by 4,"},{"Start":"00:56.990 ","End":"01:00.020","Text":"that equals to 90 divided by 4,"},{"Start":"01:00.020 ","End":"01:02.845","Text":"and that equals to 22.5."},{"Start":"01:02.845 ","End":"01:11.030","Text":"I\u0027m looking for the category where I have the 22.5 person."},{"Start":"01:11.030 ","End":"01:12.680","Text":"Well, in order to find this,"},{"Start":"01:12.680 ","End":"01:17.465","Text":"let\u0027s first calculate the cumulative frequency that\u0027s big F right here."},{"Start":"01:17.465 ","End":"01:19.940","Text":"Well, here in this category, that\u0027ll be 10."},{"Start":"01:19.940 ","End":"01:22.805","Text":"Here that\u0027ll be 10 plus 20, that\u0027ll be 30,"},{"Start":"01:22.805 ","End":"01:24.835","Text":"30 that\u0027ll be 60,"},{"Start":"01:24.835 ","End":"01:28.200","Text":"that\u0027ll be 80 and that\u0027ll be 90."},{"Start":"01:28.200 ","End":"01:32.625","Text":"I\u0027m looking for the 22.5 person."},{"Start":"01:32.625 ","End":"01:35.351","Text":"So it\u0027s not in this first category,"},{"Start":"01:35.351 ","End":"01:38.975","Text":"it\u0027s in this category because up to here,"},{"Start":"01:38.975 ","End":"01:40.370","Text":"I have 30 people,"},{"Start":"01:40.370 ","End":"01:44.135","Text":"looking for the place of this person right here."},{"Start":"01:44.135 ","End":"01:49.380","Text":"This would be the category of the first quartile."},{"Start":"01:49.380 ","End":"01:53.540","Text":"What about the location of the third quartile?"},{"Start":"01:53.540 ","End":"01:56.975","Text":"That will be 3n divided by 4."},{"Start":"01:56.975 ","End":"02:01.805","Text":"That equals to 90 or 3 times 90 divided by 4,"},{"Start":"02:01.805 ","End":"02:06.080","Text":"and that equals to 67.5."},{"Start":"02:06.080 ","End":"02:08.225","Text":"Now, where is that located?"},{"Start":"02:08.225 ","End":"02:10.310","Text":"Well, up to the third category we have"},{"Start":"02:10.310 ","End":"02:14.390","Text":"60 people and up to the fourth category we have 80 people,"},{"Start":"02:14.390 ","End":"02:23.330","Text":"so the person in the 67.5 position would be in this category right here."},{"Start":"02:23.330 ","End":"02:28.100","Text":"This then would be the category for the first quartile,"},{"Start":"02:28.100 ","End":"02:31.040","Text":"and this would be the category for the third quartile."},{"Start":"02:31.040 ","End":"02:37.805","Text":"Excellent. How do we calculate the value for the first quartile, Q_1?"},{"Start":"02:37.805 ","End":"02:45.470","Text":"Well, that equals to L_0 plus n divided by 4 minus big F of x"},{"Start":"02:45.470 ","End":"02:53.460","Text":"at n minus 1 divided by small f of x at n times L_1 minus L_0."},{"Start":"02:53.460 ","End":"02:56.720","Text":"That\u0027s the equation."},{"Start":"02:56.720 ","End":"02:58.820","Text":"Now, what does that equal to?"},{"Start":"02:58.820 ","End":"03:04.250","Text":"Well, L_0 is the lower bound of the category where we found the first quartile,"},{"Start":"03:04.250 ","End":"03:08.055","Text":"That\u0027s 45 plus, now,"},{"Start":"03:08.055 ","End":"03:09.480","Text":"what\u0027s n divided by 4?"},{"Start":"03:09.480 ","End":"03:10.680","Text":"We would calculate that,"},{"Start":"03:10.680 ","End":"03:17.840","Text":"that\u0027s 22.5 minus the cumulative frequency of the preceding category."},{"Start":"03:17.840 ","End":"03:19.790","Text":"Well, this is a category for Q_1,"},{"Start":"03:19.790 ","End":"03:22.130","Text":"the preceding category is right here."},{"Start":"03:22.130 ","End":"03:24.740","Text":"So we\u0027re looking at this value right here."},{"Start":"03:24.740 ","End":"03:28.535","Text":"That\u0027s a cumulative frequency of the preceding category."},{"Start":"03:28.535 ","End":"03:35.900","Text":"So that the minus 10 divided by the frequency of this category that\u0027s 20."},{"Start":"03:35.900 ","End":"03:38.300","Text":"Well, what\u0027s the width here?"},{"Start":"03:38.300 ","End":"03:41.165","Text":"That\u0027ll be 15 minus 45."},{"Start":"03:41.165 ","End":"03:49.775","Text":"Now, all this comes out to 48.125 kilograms."},{"Start":"03:49.775 ","End":"03:52.845","Text":"Now what about Q_3?"},{"Start":"03:52.845 ","End":"03:58.560","Text":"Well, again, that\u0027s L_0 plus 3n and divided by"},{"Start":"03:58.560 ","End":"04:06.230","Text":"4 minus big F of x at the previous category,"},{"Start":"04:06.230 ","End":"04:12.890","Text":"divided by the frequency times the width of the category."},{"Start":"04:12.890 ","End":"04:16.700","Text":"Well, what\u0027s the category for Q_3?"},{"Start":"04:16.700 ","End":"04:19.760","Text":"Well, that\u0027s 60, that\u0027s a lower bound right here."},{"Start":"04:19.760 ","End":"04:22.220","Text":"So that\u0027ll be 60 plus,"},{"Start":"04:22.220 ","End":"04:23.750","Text":"now 3n divided by 4."},{"Start":"04:23.750 ","End":"04:26.005","Text":"Well, that\u0027s 67.5."},{"Start":"04:26.005 ","End":"04:33.545","Text":"67.5 minus the cumulative frequency of the previous category."},{"Start":"04:33.545 ","End":"04:36.200","Text":"Well, that\u0027s this thing right here."},{"Start":"04:36.200 ","End":"04:41.013","Text":"That\u0027ll be 60 divided by the frequency of this category right here"},{"Start":"04:41.013 ","End":"04:46.040","Text":"that\u0027s 20 times 65 minus 60,"},{"Start":"04:46.040 ","End":"04:48.020","Text":"that\u0027s the width of this category."},{"Start":"04:48.020 ","End":"04:58.355","Text":"That equals to 61.875 kilograms. That\u0027s great."},{"Start":"04:58.355 ","End":"05:05.015","Text":"Now let\u0027s calculate the IQR or the Inter-quartile Range."},{"Start":"05:05.015 ","End":"05:09.725","Text":"The IQR or the Inter-quartile Range,"},{"Start":"05:09.725 ","End":"05:14.030","Text":"that\u0027s equals to Q_3 minus Q_1."},{"Start":"05:14.030 ","End":"05:18.315","Text":"Well, that\u0027s 61.875,"},{"Start":"05:18.315 ","End":"05:25.410","Text":"that\u0027s Q_3 minus 48.125, that\u0027s Q_1."},{"Start":"05:25.410 ","End":"05:30.450","Text":"That equals to 13.75."},{"Start":"05:30.450 ","End":"05:32.340","Text":"What does this mean?"},{"Start":"05:32.340 ","End":"05:35.270","Text":"That\u0027s in kilograms first of all, that\u0027s the units."},{"Start":"05:35.270 ","End":"05:36.485","Text":"What does this mean?"},{"Start":"05:36.485 ","End":"05:42.370","Text":"That means that 50 percent of the central population"},{"Start":"05:42.370 ","End":"05:49.380","Text":"are located in a middle range of 13.75 kilograms."}],"ID":13369},{"Watched":false,"Name":"Exercise 2","Duration":"6m 33s","ChapterTopicVideoID":12853,"CourseChapterTopicPlaylistID":265043,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12853.jpeg","UploadDate":"2018-07-30T17:58:05.2200000","DurationForVideoObject":"PT6M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this question, we\u0027re given a histogram that displays"},{"Start":"00:02.970 ","End":"00:06.615","Text":"the probability distribution of height in centimeters for a certain group,"},{"Start":"00:06.615 ","End":"00:09.600","Text":"and we\u0027re asked to find the inter-quartile range."},{"Start":"00:09.600 ","End":"00:11.685","Text":"Here we have the histogram."},{"Start":"00:11.685 ","End":"00:16.035","Text":"Here on the x-axis we have the height in categories,"},{"Start":"00:16.035 ","End":"00:20.505","Text":"and on the y-axis we have the density and why density?"},{"Start":"00:20.505 ","End":"00:22.079","Text":"Well, this is a histogram."},{"Start":"00:22.079 ","End":"00:25.620","Text":"Now, in order to calculate the inter-quartile range,"},{"Start":"00:25.620 ","End":"00:28.740","Text":"the first thing that we want to do is we want to convert this"},{"Start":"00:28.740 ","End":"00:32.940","Text":"into a frequency table by categories, so let\u0027s do that."},{"Start":"00:32.940 ","End":"00:35.210","Text":"Here\u0027s a frequency table,"},{"Start":"00:35.210 ","End":"00:36.905","Text":"and we have our categories."},{"Start":"00:36.905 ","End":"00:39.110","Text":"Let\u0027s say from 155-160,"},{"Start":"00:39.110 ","End":"00:40.835","Text":"that\u0027s this category right here,"},{"Start":"00:40.835 ","End":"00:43.700","Text":"160-165 that\u0027s this category."},{"Start":"00:43.700 ","End":"00:46.550","Text":"The next category is 165-117,"},{"Start":"00:46.550 ","End":"00:48.125","Text":"and so on and so forth."},{"Start":"00:48.125 ","End":"00:50.570","Text":"Now, the next thing that I want to do,"},{"Start":"00:50.570 ","End":"00:52.250","Text":"is we want to calculate the frequency,"},{"Start":"00:52.250 ","End":"00:53.510","Text":"but here we have density,"},{"Start":"00:53.510 ","End":"00:54.965","Text":"so how do we do that?"},{"Start":"00:54.965 ","End":"00:56.960","Text":"Well, we have here the key."},{"Start":"00:56.960 ","End":"01:02.300","Text":"We know that 1 square equals to 5 people,"},{"Start":"01:02.300 ","End":"01:05.450","Text":"so how many squares do I have in the first category?"},{"Start":"01:05.450 ","End":"01:08.255","Text":"Well, I have only 1 square right here,"},{"Start":"01:08.255 ","End":"01:11.020","Text":"so 1 times 5, that will be 5."},{"Start":"01:11.020 ","End":"01:13.520","Text":"How many squares do I have in this category?"},{"Start":"01:13.520 ","End":"01:14.780","Text":"Well, that\u0027s 1, 2."},{"Start":"01:14.780 ","End":"01:16.805","Text":"2 times 5 is 10,"},{"Start":"01:16.805 ","End":"01:18.110","Text":"and then this category,"},{"Start":"01:18.110 ","End":"01:19.895","Text":"they\u0027ll be 1, 2, 3,"},{"Start":"01:19.895 ","End":"01:22.265","Text":"that would be 15, 3 times 5."},{"Start":"01:22.265 ","End":"01:23.600","Text":"In the next category,"},{"Start":"01:23.600 ","End":"01:25.040","Text":"that will be 1, 2, 3, 4,"},{"Start":"01:25.040 ","End":"01:26.660","Text":"5, 6, 7, 8,"},{"Start":"01:26.660 ","End":"01:27.950","Text":"8 times 5,"},{"Start":"01:27.950 ","End":"01:31.580","Text":"that\u0027s 40, and in the last category, that\u0027s 1,"},{"Start":"01:31.580 ","End":"01:33.050","Text":"2, 3, 4, 5, 6,"},{"Start":"01:33.050 ","End":"01:35.645","Text":"6 times 5 that\u0027s 30."},{"Start":"01:35.645 ","End":"01:38.265","Text":"So n here when we sum all this up,"},{"Start":"01:38.265 ","End":"01:40.115","Text":"that will be equal to a 100."},{"Start":"01:40.115 ","End":"01:44.390","Text":"Now, also let\u0027s calculate the cumulative frequency."},{"Start":"01:44.390 ","End":"01:45.740","Text":"So that\u0027ll be 5,"},{"Start":"01:45.740 ","End":"01:48.770","Text":"5 plus 10, that\u0027s 15, 15 plus 15,"},{"Start":"01:48.770 ","End":"01:50.900","Text":"that\u0027s 30, 30 plus 40,"},{"Start":"01:50.900 ","End":"01:54.390","Text":"that\u0027s 70, and 70 plus 30, that\u0027s a 100."},{"Start":"01:54.470 ","End":"02:01.710","Text":"Having done that, let\u0027s now find the location of Q_1 and Q_3,"},{"Start":"02:01.710 ","End":"02:05.165","Text":"and where are they situated in the categories."},{"Start":"02:05.165 ","End":"02:08.557","Text":"Well, for Q_1, well,"},{"Start":"02:08.557 ","End":"02:12.830","Text":"that would be the n divided by fourth place."},{"Start":"02:12.830 ","End":"02:17.890","Text":"In our case, so that\u0027ll be a 100 divided by that equals to 25,"},{"Start":"02:17.890 ","End":"02:23.919","Text":"so Q_1 is the 25th person."},{"Start":"02:23.919 ","End":"02:26.215","Text":"Now, where\u0027s the 25th person?"},{"Start":"02:26.215 ","End":"02:27.770","Text":"Well, not here and not here,"},{"Start":"02:27.770 ","End":"02:30.810","Text":"because here up to the second category,"},{"Start":"02:30.810 ","End":"02:32.880","Text":"we have only 15 people."},{"Start":"02:32.880 ","End":"02:34.955","Text":"Just in the third category,"},{"Start":"02:34.955 ","End":"02:36.520","Text":"we go up to 30 people,"},{"Start":"02:36.520 ","End":"02:41.390","Text":"so the 25th person is in this category right here."},{"Start":"02:41.390 ","End":"02:44.170","Text":"What about Q_3?"},{"Start":"02:44.170 ","End":"02:47.535","Text":"Well, that\u0027s 3n divided by 4,"},{"Start":"02:47.535 ","End":"02:53.265","Text":"that equals to 300 divided by 4 and that equals to 75."},{"Start":"02:53.265 ","End":"02:55.365","Text":"Where is the 75th person?"},{"Start":"02:55.365 ","End":"02:58.410","Text":"The fourth category goes up to 70th person,"},{"Start":"02:58.410 ","End":"03:04.105","Text":"so it\u0027s only in the last category that we have the 75th person."},{"Start":"03:04.105 ","End":"03:06.815","Text":"So having done that,"},{"Start":"03:06.815 ","End":"03:11.255","Text":"let\u0027s now calculate the values of Q_1 and Q_3."},{"Start":"03:11.255 ","End":"03:13.760","Text":"Well, how do we calculate Q_1?"},{"Start":"03:13.760 ","End":"03:22.050","Text":"That equals to L_0 plus n divided by 4 minus f of x,"},{"Start":"03:22.050 ","End":"03:31.945","Text":"F of X_M minus 1 divided by the frequency of X_m times L_1 minus L_0."},{"Start":"03:31.945 ","End":"03:34.460","Text":"Now, what does that equal to?"},{"Start":"03:34.460 ","End":"03:38.050","Text":"Well, let\u0027s take a look at the category of Q_1,"},{"Start":"03:38.050 ","End":"03:39.620","Text":"that\u0027s this thing right here."},{"Start":"03:39.620 ","End":"03:41.450","Text":"So what\u0027s the lower bound here?"},{"Start":"03:41.450 ","End":"03:47.690","Text":"That\u0027s 165, so that\u0027s 165 plus now n divided by 4,"},{"Start":"03:47.690 ","End":"03:55.070","Text":"that\u0027s 25 minus what\u0027s the cumulative frequency of the preceding category?"},{"Start":"03:55.070 ","End":"03:57.110","Text":"Well, this is the category of Q_1,"},{"Start":"03:57.110 ","End":"03:59.465","Text":"the preceding categories here,"},{"Start":"03:59.465 ","End":"04:02.330","Text":"and the cumulative distribution is right here,"},{"Start":"04:02.330 ","End":"04:05.630","Text":"that\u0027s 15 divided by"},{"Start":"04:05.630 ","End":"04:12.589","Text":"the frequency of the category where Q_1 is located,"},{"Start":"04:12.589 ","End":"04:17.360","Text":"and that\u0027s right here, that\u0027s 15 times L_1."},{"Start":"04:17.360 ","End":"04:23.495","Text":"Well, that\u0027s 170. That\u0027s the upper bound minus the lower bound, 165."},{"Start":"04:23.495 ","End":"04:25.520","Text":"That\u0027s basically the width of the category."},{"Start":"04:25.520 ","End":"04:34.590","Text":"Now, that equals to 1681 and 1/3 centimeters."},{"Start":"04:34.590 ","End":"04:39.890","Text":"So likewise, we\u0027ll do the same thing for Q_3, so here we go."},{"Start":"04:39.890 ","End":"04:45.120","Text":"Q_3 now is equal to L_0 plus,"},{"Start":"04:45.120 ","End":"04:52.035","Text":"now it\u0027ll be 3n divided by 4 minus F of X_M minus 1,"},{"Start":"04:52.035 ","End":"04:59.110","Text":"divided by f of x_m times L_1 minus L_0."},{"Start":"04:59.110 ","End":"05:06.210","Text":"Now here, what\u0027s the lower bound of the category where Q_3 is located?"},{"Start":"05:06.210 ","End":"05:08.150","Text":"Well, that\u0027s this category right here."},{"Start":"05:08.150 ","End":"05:14.540","Text":"So the lower bound is 180 plus 3n divided by 4."},{"Start":"05:14.540 ","End":"05:15.770","Text":"We\u0027ve calculated that,"},{"Start":"05:15.770 ","End":"05:22.010","Text":"that\u0027s 75 minus the cumulative frequency of the preceding category."},{"Start":"05:22.010 ","End":"05:25.610","Text":"Well, here\u0027s the preceding category and the cumulative frequency is"},{"Start":"05:25.610 ","End":"05:31.610","Text":"70 divided by the frequency of the category,"},{"Start":"05:31.610 ","End":"05:36.350","Text":"that\u0027ll be 30 times the width of the category."},{"Start":"05:36.350 ","End":"05:40.175","Text":"Well, that\u0027s 190 minus 180,"},{"Start":"05:40.175 ","End":"05:46.845","Text":"and all that comes to 181 and 2/3 centimeters."},{"Start":"05:46.845 ","End":"05:51.380","Text":"Excellent. Now we have Q_1 and Q_3,"},{"Start":"05:51.380 ","End":"05:55.190","Text":"so how do we calculate the inter-quartile range?"},{"Start":"05:55.190 ","End":"06:01.385","Text":"The inter-quartile range, that equals to Q_3 minus Q_1,"},{"Start":"06:01.385 ","End":"06:10.280","Text":"and that equals to 181 and 2/3 minus 168 and 1/3,"},{"Start":"06:10.280 ","End":"06:16.345","Text":"and that equals to 13 and 1/3 centimeters."},{"Start":"06:16.345 ","End":"06:20.405","Text":"So this is the inter-quartile range."},{"Start":"06:20.405 ","End":"06:27.170","Text":"That\u0027s where 50 percent of the middle population,"},{"Start":"06:27.170 ","End":"06:28.610","Text":"that\u0027s where it\u0027s situated,"},{"Start":"06:28.610 ","End":"06:33.600","Text":"in a range of 13 and 1/3 centimeters."}],"ID":13370}],"Thumbnail":null,"ID":265043}]

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