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GRE Brain Twister: Non-Standard Deviation (Answer)

Here’s the answer to this week’s GRE Brain Twister!

Set A consists of five positive numbers. SAT B consists of the square roots of each of the five numbers from Set A. If the standard deviation of Set B > standard deviation of Set A, which of the following must be true?

I. The range of the numbers in Set A is greater than 1

II. At least one of the numbers in Set A is less than 1

III. The range of Set A > Set B

(A) I only

(B) III only

(C) I and III

(D) I and II

(E) II only

Explanation:

The best way to start this question is by noticing the fact that the standard deviation of Set A increases when you take the square root of each of the five numbers. This is strange because when you take the square root of a positive number that number decreases:

4, 9, 16, 25, 36

2, 3, 4, 5, 6

We can see that the standard deviation of the second group of numbers, which is square root of the corresponding number above, is much lower (we don’t actually need to figure out the exact standard deviations by using that tedious formula).

So how is it that the standard deviation of Set B above is greater than Set A? The only way for this to happen is if at least some of the numbers in Set A are fractions between 0 and 1. For instance, here are some possible values:

Set A: 1/36, 1/25, 1/16, 1/9, ¼

Set B: 1/6, 1/5, ¼, ½

Rarely when standard deviation comes up will the test expect you to calculate it. Instead, you should have a sense of when the standard deviation of one group of numbers is greater than another. One way to determine this is range, the difference between the least and the greatest number.

Range of Set A: ¼ – 1/36 = 5/36

Range of Set B: 3/6 – 1/6 = 1/3

Another way to determine if the standard deviation of one group of numbers is greater than another, when the range is the same, is if numbers “clump” around the mean—the more such clumping the lower the standard deviation. Here, we have a greater range in set B. The spread of the numbers in the two sets isn’t too much different. Therefore, we can conclude that Set B has the greater standard deviation. The insight: at least one of the integers in Set A must be a fraction between 0 and 1. (Roman numeral II).

Plugging in the numbers we used above, we can disprove (A):

Set A: 1/36, 1/25, 1/16, 1/9, ¼

Set B: 1/6, 1/5, ¼, ½

Finally the third roman numeral, we can see (using the numbers above) that Set B actually has the larger range. Therefore, III is untrue, leaving us with answer (E).