Measures of Central Tendency

Transcript

0:00Introduction

Hello everyone! When someone says the average student in a class did well, what does average mean? Is it all the marks added up and divided by the number of students? The marks of the student exactly in the middle? Or the marks that most students got? Here is the surprise: all three are fair answers, and for the same data they can give different numbers! They are the mean, the median and the mode: the measures of central tendency. They come up often in JEE, so let's begin.

0:32Mean, median and mode

So, first the mean. We write the observations as x i. The small i below the x is a subscript, and it just numbers them, okay? The mean is x bar, an x with a little bar on top, and the big Greek letter, capital sigma, means add them all up. So x bar equals one over n, times the sum of the x i, where n is the number of observations. Clear? Now, if a value appears many times, we multiply it by its frequency, f i. So for a frequency table, x bar equals the sum of f i times x i, over the sum of f i. Notice the bottom: we divide by the total frequency, not by the number of rows. Right? Next, the median. It is the middle value, but only after arranging the data in order. Many students forget that step, okay? When n is odd, there is one middle value. But when n is even, which value is the middle? There is no single middle value, very good! So we take the two middle values and average them. That's it. And the mode, the easiest of the three, is simply the value that occurs most often. Got it?

1:46Grouped data: mean by step deviation

Now, grouped data, where the data comes in classes. Capital N is the total frequency, small h is the class width, and x i, the class mark, is the middle point of each class. With me? Class marks can be messy numbers, so we shrink them. Pick a convenient value, capital A, often the middle class mark. Then u i equals x i minus capital A, over h. This is called step deviation. Let me ask you this: with equal class widths, what kind of numbers are the u i? Small whole numbers, yes, exactly! When capital A is one of the class marks, they are minus two, minus one, zero, one, two, and the dots mean the pattern goes on, okay? Then we undo the change. x bar equals capital A, plus h times the sum of f i u i, over capital N, the total frequency. Be careful, because a very common slip is forgetting to multiply by h. Right?

2:49Grouped data: median and mode

Now the median of grouped data. The cumulative frequency is the running total of the frequencies down the table, and the median class is where it first reaches N over two. Okay? Then the median equals l, plus, N over two minus capital C, over f, all times h. It looks heavy, but it is really simple. Clear? Small l is the lower limit of the median class, and small f is its frequency. Now the tricky one. Does capital C include the median class, or only the classes before it? Only the classes before it, very good! This is the classic trap here, and our second example will test it. Got it? Now the mode of grouped data. Mode equals l, plus, f one minus f zero, over two times f one, minus f zero, minus f two, all times h. Here l is the lower limit of the modal class, okay? The modal class is the one with the highest frequency, f one. Then f zero and f two are the frequencies of its neighbours, the classes just before and just after it. With me?

3:59Three handy results

Alright, three handy results. First, for moderately skewed data, mode is approximately three times the median, minus two times the mean. This wavy equals sign means approximately equal, so it gives only an estimate. Clear? Next, two groups, each with its own mean. What is the mean of both groups together? Just average the two means, right? No, no, wait, the bigger group must count more! So the combined x bar equals n one times x bar one, plus n two times x bar two, all over n one plus n two, where n one and n two are the group sizes. And if every x i becomes a times x i, plus b, what happens to the mean? It changes in exactly the same way! y bar equals a times x bar, plus b, and the double arrow on screen means implies. Got it? And the median and the mode change the same way too. No need to start again from scratch, right?

5:02Example 1: mean and median of a frequency table

Now example one, an easy one. Find the mean and median of this distribution. The values of x are one, two, three, four and five, with frequencies three, five, eight, three and one. Pause the video and try it first. Done? Okay, let us check. The frequencies add up to twenty. Each f times x gives three, ten, twenty four, twelve and five, which add up to fifty four. With me? So the mean is fifty four over what? There are five values of x in the table, so over five, right? No, no, careful, never over five! We divide by the total frequency, twenty. Fifty four over twenty gives two point seven, and that is the mean. Clear? Now the median. Twenty is an even number, so we need the tenth and the eleventh values. The cumulative frequencies are three, eight, sixteen, nineteen and twenty. So where do the tenth and eleventh values sit? Yes, exactly! The running total is eight up to the value two, and it jumps to sixteen with the value three. So both are three, and the median is three. See, two different centres for the same data, just like our opening question!

6:22Example 2: grouped median and the cumulative frequency trap

Now a harder one. The classes are zero to ten, ten to twenty, and so on up to forty to fifty. The little dash between two numbers is read as the word to. The frequencies are five, eight, twelve, ten and five. Find the median. Pause the video and try it first, and watch out for capital C, okay? Ready? Let us work it out. The total frequency, capital N, is forty, so N over two is twenty. Clear? The running totals are five, thirteen, twenty five, thirty five and forty. So where does the running total first reach twenty? In the class twenty to thirty, very good! That is the median class, so l equals twenty, f equals twelve, and h equals ten. Now comes the trap: is capital C twenty five, or thirteen? It is thirteen, not twenty five! Twenty five already includes the median class, and capital C counts only the classes before it. Got it? Now just put it in. The median equals twenty, plus, twenty minus thirteen, over twelve, times ten. That is twenty plus thirty five over six, which is about twenty five point eight three. Right? For this same data, other examples on this page find the mean, twenty five point five, and the mode, about twenty six point six seven. Three measures, three different numbers. That answers our opening question!

7:56Quick revision: formulas, median trap, empirical relation

So, quick revision. The mean is the sum of f i times x i, over the sum of f i; the median is the middle value; and the mode is the most frequent. In the grouped median, capital C counts only the classes before the median class. Mode is about three times the median minus two times the mean, and a times x, plus b, moves all three. Now try the practice problems on this page. All the best!