JEE Advanced, AIEEE and JEE Main (2013–2018) Questions: Statistics

Transcript

0:00Introduction

Hello everyone! Let me start with a small puzzle. Someone has a set of observations. The sum of their squares is four hundred, and their plain sum is eighty. Now, how many observations could there be? Could there be nine of them, or twelve, or fifteen, or eighteen? Only one of these four is possible. So which one, and why? We will settle it with a real past-paper question in Example two. Today we work through past entrance-exam questions on statistics, from the older papers and from JEE Main. These papers often come back to a few key ideas. So, let's begin.

0:38Idea 1: the working formula for variance

So, idea number one: the variance. The Greek letter sigma is the standard deviation, so sigma squared, with the small two up top, is the variance. The letter x with a bar on top, x bar, is the mean, and x sub i, with the small i written low, is the observation number i. The big capital sigma means the sum of. So sigma squared equals one over n, times the sum of, x sub i minus x bar, the whole squared. And we divide by n, the number of observations, as our course does. Okay? Now, in past papers this form is usually slow to use. So use the working formula: sigma squared equals the sum of x sub i squared, over n, minus x bar squared. Right? In words: the mean of the squares, minus the square of the mean. Many questions in this set need nothing more. Clear? Now let me ask you this: can a variance ever be negative? No, never, and this is the key! The variance is an average of squares, so it is zero or more. The sign like a greater-than with a line under it means greater than or equal to, and the double arrow means it follows that. So the sum of x sub i squared is at least one over n, times, in big brackets, the sum of x sub i, the whole squared. Keep this line in mind, because it settles our puzzle in Example two. With me?

2:07Idea 2: shift and scale

Idea number two: shift and scale. 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. Then the mean follows the same rule: y bar equals a times x bar, plus b. Okay? Now for the spread. The small x and y written below sigma tell you which data set. The variance gets multiplied by a squared: sigma sub y squared equals a squared times sigma sub x squared. So for the standard deviation, sigma sub y equals mod a times sigma sub x. The two vertical bars are the modulus, mod a, because a standard deviation is never negative. Got it? Now, what about the plus b part? What does it do to the variance? It does nothing at all, because adding b moves every value by the same amount, so every distance from the mean stays the same. So adding a constant never changes the spread. Right? Now try a real past-paper question. Every student in a class is given ten grace marks. Which of the mean, the median, the mode and the variance does not change? Yes, exactly, the variance! The plus ten on screen is the grace marks. The mean, the median and the mode all move up by ten, but the spread stays exactly where it was. Clear? Here is one more from the same papers. Population A is one hundred and one, up to two hundred, and population B is one hundred and fifty one, up to two hundred and fifty. The three dots mean and so on. Population B is just population A, shifted by fifty. Here V stands for variance, so the ratio of the two variances, V sub A over V sub B, is simply one, with no calculation at all! With me?

4:04Idea 3: evenly spaced data

Idea number three: ready-made results for evenly spaced data. For the first n natural numbers, one, two, up to n, the variance is n squared minus one, over twelve. Learn this one by heart. Okay? Now take an arithmetic progression, where each term goes up by a common difference d. What is its variance? Use idea number two. Yes, shift and scale again! On screen, A P is short for arithmetic progression. Its terms are d times one, two, up to n, plus a fixed constant. The constant changes nothing, and the factor d multiplies the variance by d squared. So sigma squared equals d squared times, n squared minus one, over twelve. Got it?

4:52Example 1: variance of the first 50 even natural numbers

Let's start with an easy one, from JEE Main, two thousand fourteen. Find the variance of the first fifty even natural numbers. Pause the video and try it first. Done? Okay, let's check. The numbers are two, four, six, up to one hundred. Each one is two times one of the numbers one, two, three, up to fifty. The cross means times. Right? For one up to fifty, idea number three gives the variance directly: fifty squared minus one, over twelve. Fifty squared is two thousand five hundred, so that is two thousand four hundred and ninety nine, over twelve. Okay? Now, every number is doubled. So the variance also doubles, right? No, no, wait, that is the trap! Scaling by two multiplies the variance by two squared, which is four. The dot means times, so sigma squared is four times two thousand four hundred and ninety nine, over twelve. Got it? Four over twelve is one third, so this is two thousand four hundred and ninety nine over three, which gives eight hundred and thirty three. That's it, two lines of work!

6:05Example 2: find n, since variance is never negative

Now, back to our puzzle, a real past-paper question. For n observations, the sum of x sub i squared is four hundred, and the sum of x sub i is eighty. Which of nine, twelve, fifteen and eighteen can be n? Pause the video and try it first. Done? Okay, let's check. The mean, x bar, is the sum over n, so eighty over n. Now use the working formula. Sigma squared equals four hundred over n, minus the square of eighty over n, and that square is six thousand four hundred over n squared. Right? Now let's just try the first option, n equal to nine. The mean is eighty over nine, a perfectly fine number. So nine is possible, right? Oh no, careful, look at the variance! Four hundred over nine, minus six thousand four hundred over eighty one, gives minus two thousand eight hundred over eighty one. The small arrowhead pointing left means less than, so this is less than zero, a negative variance. That is impossible, so nine is out, and the mean alone never warns you. Clear? So, instead of trying each option, use idea number one just once. Sigma squared is at least zero, so four hundred over n is at least six thousand four hundred over n squared. Multiply both sides by n squared, which is positive, and four hundred times n is at least six thousand four hundred. With me? Divide both sides by four hundred, and n is at least sixteen. So of nine, twelve, fifteen and eighteen, only eighteen works. And our puzzle is solved: the other three options would need a negative variance, and no data can ever do that. Got it?

7:54Quick revision: working formula, shift and scale, A.P. data

So, quick revision. First, the variance is the mean of the squares, minus the square of the mean, and it is never negative. Second, adding a constant leaves the variance unchanged, and multiplying the data by a constant a multiplies it by a squared. Third, for data in an arithmetic progression, sigma squared equals d squared times, n squared minus one, over twelve. Now go and try the practice problem on this page. All the best!