Changing Set, Changing Standard Deviation

October 5, 2010

Standard deviation is one of the most feared topics on the GMAT Quantitative section. It doesn't need to be.

I wrote an introduction to GMAT standard deviation some time ago, and if you are new to the concept, I suggest you click over and read that article now. Today I'm going to focus on a couple of more advanced concepts.

When Sets Change

For today's purposes, let's use a set consisting of four elements: {10, 11, 13, 20}. That set has a median of 12 and a mean of 13.5. We don't need to know how to calculate standard deviation, so let's call its standard deviation x.

The standard deviation of a set measures the distance between the average term in the set and the mean. So, if the numbers get closer to the mean, the standard deviation gets smaller. If the numbers get bigger, the reverse happens.

Given this concept and the set {10, 11, 13, 20}, try your hand at a quick quiz. For each of the following changes to the set, does the standard deviation of the set increase, decrease, or stay the same?

1. The smallest term increases by 1.
2. The largest term increases by 1.
3. Every term increases by 1.
4. Every term decreases by 1.
5. Every term is doubled.
6. Every term is divided by 2.

Answers

Let's see how you did.

1. When the smallest term increases by 1, it gets closer to the mean. Thus, the average distance from the mean gets smaller, so the standard deviation decreases.
2. When the largest term increases by 1, it gets farther from the mean. Thus, the average distance from the mean gets bigger, so the standard deviation increases.
3. When each term moves by the same amount, the distances between terms stays the same. The mean moves up to 14.5, but the distances don't change, meaning that the standard deviation stays the same.
4. Same as the previous example--stays the same.
5. Be careful with this one. It seems similar to the previous two, but when every term is doubled, the distance between each term doubles, as well. The set becomes {20, 22, 26, 40} with a mean of 27. Since the terms are farther apart, the standard deviation increases.
6. The logic here is the opposite of the previous item. When every term is halved, the distance between the numbers gets smaller: {5, 5.5, 6.5, 10} with a mean of 6.75. The standard deviation decreases.

These six examples don't comprise every possible variation the GMAT could throw at you, but anything not on this list can be solved using the same concepts. Think about the distance between terms, and as always, avoid time-consuming calculations.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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