Measures of Dispersion

Transcript

0:00Introduction

Hello everyone! Let me start with a small puzzle. Take the four numbers one, two, three and ten. We want to know how spread out they are, so we measure their average distance from a centre. But which centre should we use, the mean or the median? Which one gives the smaller average distance? Most students say the mean. Hold on to your guess; by the end of this video, you will know for sure. Spread, or dispersion, comes up often in JEE statistics, so let's begin.

0:32Mean deviation: average distance from a centre

So, how do we measure spread? We average the distances of the observations from some centre, which we call a, okay? Here x sub i, with the small i below, is observation number i, and n is how many there are. The big sigma sign means add them all up, and the two vertical bars are the modulus. So M D of a, the mean deviation about a, equals one over n, times the sum of mod x sub i minus a, okay? Now let me ask you this: why do we need the modulus bars at all? Why not just add the plain differences, x sub i minus a? Yes, exactly, because some differences are negative and some are positive, and they would cancel each other! The modulus turns every difference into a distance, never negative. And for the centre a, we take the mean, x bar, the x with a bar on top, or the median. Clear? For a frequency distribution, each value x sub i comes f sub i times, its frequency. So weight each distance by f sub i, and divide by capital N, the total frequency, which is the sum of all the f sub i. Got it?

1:48Variance and standard deviation

Now, the second way to get rid of the minus signs: square them. Sigma squared, the small Greek letter sigma with a two above it, is the variance. It is one over n, times the sum of x sub i minus x bar, the whole squared, so we divide by n, okay? And its positive square root, under the root sign, is sigma, the standard deviation, S D for short. Right? Squaring every deviation is slow, so use the working formula. Sigma squared equals the sum of x sub i squared, over n, minus x bar squared. In words, the mean of the squares, minus the square of the mean. With me? Look at the first formula again. Can the variance ever be negative? No, it can never be negative, because it is an average of squares! So the mean of the squares is always at least the square of the mean. Multiply by n, and you get this line: the sum of x sub i squared is greater than or equal to one over n, times the square of the sum of x sub i. Clear?

2:53Shift and scale

Now an area full of traps. Suppose every observation is multiplied by a constant a, and then b is added, so y sub i equals a times x sub i, plus b. First, the plus b part. Does adding a constant change the spread? No, not at all, because adding b moves every value by the same amount, so every distance stays exactly the same! The mean moves by b too, but adding a constant does not change the spread at all, okay? Now the multiplying part. Every difference gets multiplied by a. But what if a is a negative number? Does the standard deviation then become negative? No, a standard deviation can never be negative! The small x and y below sigma tell you which data set. The variance gets multiplied by a squared, so sigma y is mod a times sigma x, and the mean deviation also gets multiplied by mod a. Keeping the minus sign, or adding b to the standard deviation, is the common trap, so don't fall for it. Got it?

4:00Which centre gives the least spread?

Now, back to our puzzle about the centre. For any number a, the sum of x sub i minus a, the whole squared, equals n times sigma squared, plus n times x bar minus a, the whole squared. Okay? Look, the first term does not depend on a at all. So which value of a makes this sum the smallest? Yes, exactly, at the mean, x bar, because then the second term becomes zero! So the variance is least about the mean. Right? But for plain distances, the winner is different. The mean deviation is least about the median. Example two shows it with real numbers. With me?

4:43Topper's shortcuts

Now some results toppers simply remember. For the first n natural numbers, one, two, three, up to n, the variance is n squared minus one, over twelve. Okay? And for an A P, an arithmetic progression of n terms with common difference d, multiply that by d squared. So the variance is d squared times n squared minus one, over twelve. The first term does not appear, because adding a constant does not change the spread, remember? And the coefficient of variation, C V, is sigma over x bar, times one hundred. Between two data sets, the one with the smaller C V is more consistent. Got it?

5:24Example 1: variance and SD of 2, 4, 6, 8, 10

Now example one. Find the variance and the standard deviation of two, four, six, eight and ten. Pause the video and try it first. Done? Okay, let's check. The sum is thirty, and thirty over five gives the mean, x bar equals six. Now the squares are four, sixteen, thirty six, sixty four and one hundred, and they add up to two hundred and twenty. Okay? Two hundred and twenty over five gives forty four, the mean of the squares. Now subtract the mean, six, and the variance is thirty eight, right? No, no, careful, that is the trap! We subtract the square of the mean, six squared, which is thirty six. Forty four minus thirty six gives eight, so the variance is eight. Clear? And the standard deviation is the square root of eight, which is two root two. With me? Now the topper's check. These numbers form an A P, with d equal to two and n equal to five. So d squared is four, n squared minus one is twenty four, and four times twenty four, over twelve, gives eight again. Same answer, in one line!

6:39Example 2: mean deviation of 1, 2, 3, 10, mean vs median

Now example two, and this one settles our opening puzzle. For the numbers one, two, three and ten, compare the mean deviations about the mean and about the median. Pause the video and try it first. Done? Okay, let's check. The sum is sixteen, and sixteen over four gives a mean of four. The distances from four are three, two, one and six. Okay? They add up to twelve, and twelve over four gives three. So the mean deviation about the mean is three. Right? Now the median. There are four values, an even number of them, so what is the median? Yes, it is the average of the two middle values, two and three, so the median is two point five, not two and not three. The distances from it are one point five, zero point five, zero point five and seven point five. Clear? They add up to ten, and ten over four gives two point five. Got it? So two point five is less than three, and the median gives the smaller mean deviation, exactly as our result promised. That settles our puzzle: for the mean deviation, the median wins, not the mean.

7:53Quick revision: MD, variance shortcut, shift and scale

So, quick revision. The mean deviation is least about the median. The variance is the mean of the squares minus the square of the mean, and sigma is its square root. And when y sub i equals a times x sub i, plus b, sigma y is mod a times sigma x, and adding a constant does not change the spread. Now go and try the practice problems on this page. All the best!