How to Quickly Solve Standard Deviation Questions on the GMAT

The quantitative section of the GMAT is designed to test your understanding and application of concepts you learned in high school. The exam focuses on core mathematical concepts such as algebra, geometry and statistics. However some concepts are more engrained in the high school curriculum than others. Everyone’s done addition, multiplication, subtraction and division, but sometimes figuring out factorials or square roots may be a little more unusual.

Perhaps no concept perplexes students on the GMAT more than the standard deviation. The standard deviation (often represented by ?) is measure of dispersion around the mean. It indicates how close the numbers in a set are to the set’s average. As a simple example, the sets {5, 10, 15} and {8, 10, 12} both have the same mean (10); however they do not have the same standard deviation.

Knowing how to calculate the standard deviation is not required on the GMAT, but knowing how it’s calculated gives you a tremendous edge in answering questions about standard deviation. Calculating standard deviation is a four step process:

1)      Find the average (mean) of the set.

2)      Find the differences between each element of the set and that average.

3)      Square all the differences and take the average of the differences. This gives you the variance.

4)      Take the square root of the variance.

In this example, the average of the first set is clearly 10. The differences between the three elements are (-5, 0 and -5). Taking the square of these numbers, we get (25, 0 and 25). The average of these numbers is 50/3 or 16.67. The square root of this number will not be an integer, but it will be very close to 4. So we can assume that the standard deviation is roughly ~4 or ~4.1.

In contrast, the second set of numbers will have a much smaller standard deviation. The average is still 10, but the differences are now (-2, 0 and 2). Taking the square of these numbers, we get (4, 0 and 4). The average of these numbers is 8/3 or 2.67. The square root of 2.67 is roughly ~1.6 or ~1.7, but it’s very hard to pin down without a calculator or a lot of extra time.

This example should help highlight why the standard deviation is not explicitly calculated on the GMAT exam without a calculator: the chances of it being an integer are relatively low. However the concept it represents and the idea behind it are fair game on the test. One of the simple takeaways from the math behind the process is that, the farther the number is from the mean of the set, the more the standard deviation will increase. Specifically, the distance increases with the square of the difference, so 5 looks much farther out than 2.

This kind of concept can be tested on the GMAT, but if you know what you’re looking for, you can answer GMAT standard deviation questions very quickly. Let’s look at an example of how you might see standard deviation on the GMAT:

For the set {2, 2, 3, 3, 4, 4, 5, 5, x}, which of the following values of x will most increase the standard deviation?

(A)   1

(B)   2

(C)   3

(D)   4

(E)    5

If you recall the steps to calculating the standard deviation, what we really need to do first is to calculate the mean. (i.e. how mean are you?) You can add the eight elements together and divide by eight, but the fact that these elements follow a fairly obvious pattern helps us as well. The numbers each appear twice, and they are evenly spaced. This means that the average will be the same as the median, and the median is 3.5. Even if you take the long way, it shouldn’t take you more than 20 seconds to find that the mean of this set is 3.5

The next step is to take each element and find the difference from the mean, but this is what we need to do if the goal is to actually calculate the standard deviation. All we’re being tasked to do here is to determine which number will increase the standard deviation the most. In this regard, all we need to do is figure out which answer choice is furthest from the mean. That number will produce the biggest distance, which will then be squared and in turn produce the biggest difference in standard deviation. So although you can spend a lot of time calculating every last detail of this question, what it actually comes down to is “which of these numbers is furthest from 3.5”.

Asking about distance from a specific number is much more straightforward, and probably an elementary school level question. Yet, if you understand the concept, you can turn a GMAT standard deviation question into something a 5th grader could answer (Are you smarter than a 5th grader?). The answer is thus obviously choice A, as 1 is as far from 3.5 as possible given only these five choices.

The important thing about the standard deviation on the GMAT is that you will never have to formally calculate it, but understanding the underlying concept will help you excel at the quantitative section of the GMAT. Most standard deviation questions hinge primarily on the distance from the mean, as everything else is just a rote division or addition. Some other important concepts that will appear on GMAT standard deviation questions are:

-Standard deviation is never negative. Because standard deviation is a measure of “distance from the mean” it’s always positive, except for when…

-The standard deviation is zero whenever there is no deviation whatsoever. So if a set of data is 9, 9, 9, there’s no deviation (the mean is 9 and all terms are 9) and therefore the standard deviation is zero.

-If you multiply all terms in a set by something with an absolute value greater than 1, you will increase the standard deviation as the terms get farther apart.

-If you divide all terms in a set by something with an absolute value greater than 1, you will decrease the standard deviation as the terms get closer together.

-Multiplying the terms by -1 or adding a constant to them won’t change the standard deviation, as the terms all stay the same distance apart.

Remember: standard deviation on the GMAT is tested in a fairly qualitative way – it’s about whether you understand the concept of standard deviation and not about your capacity to grind out the calculation. So make sure you understand the concept of standard deviation – it measures generally (standard) how far data is (deviation) from the average. The more that the data gets spread farther apart, the larger the standard deviation, and the more that it gets brought closer together, the smaller the standard deviation. If you recognize these concepts, you’ll likely find that the fastest way to solve most standard deviation problems on the GMAT is by visual inspection: does the data get further apart or closer together? Standard deviation is hard to calculate without a calculator…which is why standard deviation calculations just about never appear on the GMAT. The key to GMAT standard deviation questions is a deep understanding of what standard deviation is and what it measures.

Much like taking five practice exams and getting wildly different scores, having a high variance is bad for knowing what to expect. Understanding the way standard deviations are tested on the GMAT will help you consistently get the questions right and reduce the variance of your results (hopefully with a very high mean).

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.