Lecture 3.2 - Describing Numerical Data - Mean

0:14 The next thing which we are going to discuss is how do we summarize data using numerical summaries. When we talked about the categorical variable, we introduce two descriptive measures; namely, the mode and the median. While mode was used both for nominal and ordinal type of data. We saw that if you want to summarize the data using median, it has to be a ordinal data or there should be some sense of order in the data. So, now what we are going to look at is what are the ways we can describe numerical data . So, the objective here is to actually develop measures or summaries of data which can be

1:03 used to summarize a complete data set . So, the most commonly used, so especially when we are dealing with numerical data the most commonly used descriptive measures again can be broadly categorized into two categories and these categories are one which are known as measures of central tendency and the second which are known as measures of dispersion. Now, what do we mean by measures of central tendency? Measures of central tendency as the name suggests, it talks about where the data is actually concentrated or what is the most typical value of a data set. A measure of central tendency describes or tells us

1:55 what can we expect as a typical value of a data set; whereas, the measure of dispersion also referred to as measures of variation or measures or spread talk about the variability in a data set or spread in a data set. So, we will go through these measures in detail. When we talk about measures of central tendency, the most common measure of central tendency is what we refer to as the mean of a data set. What is a mean? I define the mean of a data set to be the sum of the observations divided by the number of observations.

2:42 So, now, let us formally define what it is. Suppose, I have x 1, x 2, x n I refer to my observations as x 1, x 2, x n. For example, I have my data which is 4, 3, 1, 2 and 5. I have n equal to 5 here. My x 1 is 4; x 2 is 3; x 3 is 1; x 4 is 2 and x 5 is 1. So, in general, if I have n observations each x 1 refers to the first observation in my data set; x 2 refers to the second observation in my data set and x n refers to the nth observation

3:30 in my data set. The mean of a data set is the sum of these observations, the numerator gives me the sum of these observations which is nothing but x 1 plus x 2 plus x 3 up to x n divided by the total number of observations which is n, small n. Now, recall in when we introduce the notion of a sample and population, we said that a small n refers to what we call the sample size; whereas, capital N refers to a population size. So, if I am having a data set which is a sample,

4:20 then I do denote it or my notation is going to be small n; when I have population, when I refer to it as a population, it is going to be capital N. So, if my I can define a sample mean, when I have a sample data set to be the sum of the sample observations divided by the total number of observations, the way we have done here. I can also define a population mean which is typically refer to by the Greek alphabet mu to be x 1 plus x 2 to x n; whereas, again remember, I said n is the total number of observations in a population divided by capital N which is the total number of observations.

5:11 The definitions are the same, only thing the number of observations differ whether you are referring to a sample or whether you are referring to a population. The mean is also popularly refer to as a average. So, now, let us compute the mean for small data sets. Now, I have this data set which is given to me . So, the mean for this data set is very simple. That is x bar is 2 plus 12 plus 5 plus 7 plus 6 plus 7 plus 3. This is what I have here and that I can see is equal to 42, the numerator and which is equal to 6.

5:57 In other words, 6 is the mean of this data set. Now, let us look at another data set. See the only difference between the first data set and the second data set is in x 2 or the second observation, everything else is the same. So, I compute the mean for this data set, When I compute the mean for this data set, I find x bar is equal to 135 by 7 which is 19.285. What I want you all to observe very clearly is the difference between this data set and this data set is only the second number; but the mean difference is

6:48 very high. Now, let us look at another data set which is the following, where I the it has only 6 observations now, it does not have the last observation. The mean for this data set is going to be 21.33. We see that these two means are fairly close to each other; whereas, these two means are very different from each other . So, what this tells us is the mean even though here I have only one observation which is different. This observation and this observation are very different from each other. So, the mean is very sensitive to outliers. By outliers,

7:37 I mean what is the number which is very different from what the typical data set behaves like. So, now let us go back to another example. These are the marks obtained by 10 students in an exam . So, now, I can see that the sample mean in this case is 590 divided by 10 which is 59. So, this is the data I have . So, these are the 10 data sets. This is the data which I have, which I have just given to you 68, 79; this is the data which we have worked with ok . So, in this data, you can see that this

8:30 sum B 2 to B 11 is nothing but the sum which I have worked here. So, the sum of this is the numerator sum which is equal to 590. I have a 590 here which is the sum of all my data values and you can see the mean, the google sheet the mean gives me what is B 12 by 10 that is what is 590 divided by the total number of observations. That is what the mean gives you. Now, you can see that the google sheet has average demand, command is giving me the same average as what I have obtained before. So, average

9:25 command in the google sheet gives me the mean which we have different ok. The next thing which we are going to see is how do I obtain the mean for a grouped data. Now, if I have single. So, now, let us go back to this data which we have seen earlier which was the response from 15 individuals, we have already seen that this is the frequency 1 appears twice; 2 appears thrice; 3 appears five times; 4 appears four times and 5 appears once, this is what we have already seen. Now, for a data of this kind how do I compute the average? Now, 1 appears two times, so it appears with the frequency 2 . So, I so the way I I need to add 1 2 times. So, it is 2 into so 1 plus

10:22 1 which is giving me a 2. Similarly, 2 appears 3 times. So, the total sum of 2 is going to be 2 plus 2 plus 2 or 2 into 3 which is equal to 6. 3 appears five times, so the sum 3 contributes to its 3 into 5 which is 15; 4 into 4 which is 16 and 5 into 1 is 15. So, instead of writing 2 plus 1 plus 3 plus all of it, I can write it as 1 appears two times which contributes to 2; 2 appears three times which contributes six and this is the numerator which is summation fi xi; i going from 1 to n divided by your n or summation

11:14 fi which is equal to n. This gives me the X bar when I have discrete single value data. So, the mean here is going to be 44 divided by 15 which is equal to 2.93. [noise Now, if I go back and type in the same data in a particular sheet ok and I go back and type in the same data in a particular sheet, I add a sheet, I just type in this data .

12:01 So, the data is 1, 2, 1, 3, 4, 5 . So, I am just going to type in the data. 2, 1, 3, 4, 5, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 4, 1, 2, 3, 4, 4, 1, 2, 3, 4. I can see that there are 15 values, I just write down the average of this data set which you can see is equal

12:52 to 2.93 and this is precisely what we had got in our earlier. Now, what how do we compute the mean for group data or continuous data? Now, again recall this when I have continuous data, I do not have a particular value of the discrete data that is being taken. So, in this case recall we have what is called the midpoint of the data set or for each class interval. For example, the midpoint of the class interval 30 to 40 is 35; 40 to 50 is 45; 50 to 60 is 55. So, you look at the midpoint rather than looking at each discrete value, I know the frequency of the data set.

13:44 Now, I multiply this frequency with the midpoint, I get 105 and 6 into 45 is 270. My numerator is going to be 105 plus 270 plus 170 divided by the total number is again 50 . So, it is 2940 divided by 50 which is 58.8 . A word of caution here is this 58.8 is not the actual mean because we are approximating it only with the midpoint, I am now taking

14:29 the actual values of the data. I repeat this 58.8 is an approximation because I am multiplying the frequency with the midpoint or the best representative in this class interval which is 35. I know my data lies between 30 to 40. So, I am just saying that ok, it is around 35. So, this 35 is an approximation. So, because we are not looking at the exact data values, this average is only an approximation and not the exact value of the mean; whereas, when we looked at the discrete single value data, I took the exact value of my data and

15:17 hence, my mean matched with the exact average. This is a point to be noted. Now, how sensitive is the mean if I add a constant to every observation. Now, why are we even interested in knowing about this. For example, let us look at a case where a teacher has already given the marks to the students that is I have marks of 10 students. Now, I become benevolent, suddenly and I decide to add 5 marks to each of the students. So, this is the earlier data set which was 68 had a mean of 59. Now, I am adding 5 marks

16:11 to each of these students. So, now, the data set becomes 68 plus 5 which is 73; 79 plus 5 which is 84 and so forth 66 plus 5 which is 71. The number of observations remain the same, but I am adding a constant that is I am adding 5 to each value in my data set ok. So, what happens to the mean? So, earlier my x 1, x 2, x n were 68, 78, x 3 was 38, my x 10 was 66. I am going to add y 1 is become 73, 68 plus 5; y 2 is 84 again 78 plus y;

17:03 y 3 is 43, 38 plus 5; y 10 is 71 . So, by my definition, if I define what is x bar, x bar is going to be a summation xi divided by 10; y bar is going to be summation yi divided by 10. This is what our definitions say. So, if I add this and I compute y bar, I find out the y bar is going to be 64 by 10; 640 by 10 which is 64. You can notice that 64 is my y bar this is my x bar which is 5. So, why does this happen?

17:53 So, I know that y 1 plus y 2 plus y 3 plus y 10 by 5 or by 10 is my y bar; but y 1 is x 1 plus 5, y 2 is x 2 plus 5, y 3 is x 3 plus 5. Similarly, y 10 is also x 10 plus 5 . So, you can see that I have x 1 plus x 2 plus x 3 plus x 10 which makes y bar equal to the following . So, I have this which is nothing but x 1 plus x 2 plus x 3 plus x 10 plus 50 by 10 because this 5 is added 10 times. Now, x 1 plus x

18:51 2 plus x 10 by 10 is my x bar that is what we have already seen. x 1 plus x 2 plus x 10 is my x bar that is something which we have already seen ok. So, I have this as my x bar 50 plus 10 is 5, this is the constant. So, I have y bar which is equal to x bar plus c. So, in summary, when I add a constant to every data point what happens is the mean of the new data set is the old mean plus the same constant.

19:39 So, here what we have done in this is this is my original data set, I add a constant that is 73 every data set is a constant. This sum is equal to 640 that 640, 0 is what I have here the and the mean is 64 which is equal to 59 plus 5 . So, adding a constant to every point in the data set is what we have seen so far. The next thing is what happens to with the mean, when I multiply each observation of the data set with a constant. Again, let us look at what. So, my y x 1, x 2, x n is my

20:27 original data set, I am multiplying it with the constant. Why are we even interested in something of this kind? For example, I conduct an examination, where the marks are for 100. I conduct an examination for 100; but I want to take only 30 percent of the grade. So, the if the marks when it were for 100 has given me a particular thing, I want to take a particular percentage . So, this c could be a percentage. So, if recall I go back to the same example here, I decide to scale down each mark by 40 percent. In other sense, each mark is multiplied by my c here is 0.4 . So, you can see that

21:15 if I multiply each point 68 into 0.4 is 27.2. So, that is what I have here. I am multiplying each of my data point with 0.4; 68 into 0.4, this is 79 into 0.4, this is 66 into 0.4; I multiply each of my data set with c. Now, let us look at what is the sum of that data set is 236 giving me an average of 236 by 10 which is equal to 23.6. Now, you can notice that again, if I go back what is my y bar my y bar is y 1 plus y 2

22:10 plus my y 10 divided by 10, but my y 1 is equal to constant which times x 1 plus constant times x 2 plus constant times x n. This constant is constant for every observation. So, I can remove that constant outside and I can just have x 1 plus x 2 plus x 10 divided by 10. This is x bar; hence, y bar is constant times x bar ok. So, you can verify my constant here is 0.4; my x bar was 59, so constant times 0.4 into 59 is what I have is 23.6.

23:03 So, what we have learned when we studied about mean is we first looked at what is a definition of a mean, mean is one of the most commonly used summaries of data. We defined what was a sample mean and a population mean. But further, most of the course we are going to only deal with sample mean definition. We saw how to compute the sample mean for discrete ungrouped data and then, we looked at how to compute the sample mean for grouped data, then we saw what would happen when we manipulate data by adding a constant to each data point and by multiplying each data point with a constant. That is what we have seen

23:52 so far.