This is not a GMAT-like question at all.
The GMAT expects you to have a general understanding of what standard deviation means: it's a way of expressing how far spread out, on average, the data is from the mean. The GMAT does not expect you to know how that is calculated. The most that the GMAT expects you to know is that standard deviation is related to "variance" (see #D31 in OG2016 or 2015). You will not need to know how that relates to absolute value of difference, or sum of squares, etc.
But in any event, here's how to approach it:
What is the standard deviation of a set of numbers whose mean is 20?
(1) The absolute value of the difference of each number in the set from the mean is equal
All this is telling us is that each term is equally far from 20. Consider two scenarios:
a) Our set consists of all 20's: [20, 20, 20, 20]. The standard deviation would be 0.
b) Our set consists of mirrored pairs of numbers equally far from 20: [19, 19, 21, 21] or [-10, -10, 50, 50]. These two different sets would give us very different standard deviations.
Insufficient.
(2) The sum of the squares of the differences from the mean is greater than 100
This is the definition of variance. The standard deviation would be the square root of this number. So statement 2 tells us that the SD is less than 10, but that doesn't help us. Insufficient.
If we combine statements 1 and 2, we still have no further information on what the SD is. Insufficient.
The answer is E.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
