Official Explanation:
Standard deviation is a tricky subject. Here’s a
GMAT blog that explains absolutely everything you need to know about the standard deviation.
First we have to understand what the mean is. The mean is just the plain old average --- add up all the terms on the list, and divide by the number of terms on the list. Adding -100 and -100 would lower the mean the most, but that’s not what the question is asking.
The standard deviation is something much more complicated. There's a technical definition, which I explain in that GMAT blog, but the more informal definition is fine for this problem.
The informal definition --- standard deviation is sort of an average of the deviations from the mean. You see, every term has its own distance from the mean, and we call that a “deviation from the mean” ---- for these purposes, we count it as positive whether it’s greater than or less than the mean.
In this problem, the mean = 10. A value of 13 would have a deviation from the mean of 3. A value of 7 would also have a deviation from the mean of 3. Each of those terms, 13 and 7, are a distance of 3 from the mean, so in terms of the standard deviation, they contribute the same thing. Very high or very low numbers, far away from the mean, would have large deviations from the mean. We find the deviation from the mean of every number on the list --- all 100 numbers in this problem ---- and the standard deviation is sort of an average of all 100 of those deviations from the mean. (Again, this is not the precise definition, it’s not a strict average, but for this problem it’s close enough.)
The question gives a numerical value for the standard deviation, 4.6, but that's just a distractor. We don't need that.
The question asks us to add two numbers that lower the standard deviation the most. Well, the standard deviation is a kind of average, and if we want to lower any average, we have to add new terms that make contributions as small as possible. What's the smallest possible deviation a number could have from the mean? If we added a new term with a value of 10, that term equals the mean, so its deviation from the mean—its distance from the mean—is zero. If the new term is anything above or below 10, it will have a deviation from the mean greater than zero. Therefore, the lowest possible deviation from the mean a term can have is zero, and this is possible only if the term equals the mean of 10. Thus, if we two terms, both equal to the mean, we will be adding two terms with the lowest possible deviation from the mean, and that will lower the standard deviation, the average across all deviations from the mean, as much as possible.
Answer = E