answer to last week's MGRE Math Beast Challenge

Having gotten disgustingly sick on Monday, I never answered last week's MGRE Math Beast Challenge. Here's MGRE's explanation, which-- strangely enough-- doesn't seem to require a very deep knowledge of how standard deviations work.

The main challenge in this problem is working through the math language to figure out what the question is really asking.

First, we are given a function: f(x) = 0.27(-3.12x – 4)

Let’s distribute: f(x) = -0.8424x – 1.08

The ugliness of these numbers is a good clue that this is more of a logic problem than a straight math problem.

We are then told that Set P consists of n distinct values that are inputted into f(x). Keep in mind that n here is just the number of numbers in the set. So, if Set P were 10, 11, 12, n would simply be 3. We are also told that the values are distinct (so the set could not be 1, 1, 1, 1, 1, for instance).

Set Q consists of all the results you get from plugging the values in Set P into the function.

Let’s try a simple example. What if Set P = {1, 2}?

f(1) = -0.8424(1) – 1.08
f(1) = -1.9224

f(2) = -0.8424(2) – 1.08
(2) = - 2.7648

Thus, if Set P = {1, 2}, then Set Q = {-1.9224, -2.7648}. Note that the values in Set P are further apart (exactly 1 apart), while the values in Set Q are closer together (less than 1 apart). Thus, the standard deviation of Set P is greater. But will this always be true?

At this point the answer is either A or D. We could try other possibilities – Set P could be nearly anything, after all. But a bit of logic might prove helpful.

The standard deviation of a distinct set increases when every item in the set is multiplied by a value > 1 or < -1.

The standard deviation of a distinct set decreases when every item in the set is multiplied by a value between -1 and 1, not inclusive.

The standard deviation of a distinct set does not change when every item in the set has the same value added to it (or subtracted from it).

Thus, in the function f(x) = -0.8424x – 1.08, x undergoes two changes:

It is multiplied by a number between -1 and 1.
It has a value subtracted from it.

When you perform both these changes to every item in Set P, the first change will cause the standard deviation to decrease – that is, the numbers get closer together. The second change makes no difference to the standard deviation.

Thus, no matter what numbers you pick, Set P will always have a greater standard deviation than Set Q (said another way, running at least two distinct numbers through this particular function yields output numbers that are closer to one another than the input numbers were).

Notice that the problem specified “distinct” values? If that one word were removed from the problem, the answer would become D. Why? Without the word “distinct,” Set P could be something like {10, 10, 10}, which has a standard deviation of zero. Putting {10, 10, 10} through the function would yield {-9.504, -9.504, -9.504}, which also has a standard deviation of zero. Since it would then be possible for Quantity A to be larger but also possible for the quantities to be equal, the answer would become D. Watch out for the “distinct trap” in standard deviation problems!

The correct answer is A.

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