A certain list of 100 data has an average (arithmetic mean) of 6 and

Yes, mean won't be the same for A, B, and C but as it will differ from 6 very little then the calculations above will still hold true, so it doesn't really matter.

As for the question, GMAT SD questions are fairly straightforward and don't require actual calculation of SD, they are about the general understanding of the concept.

So we have:
A certain list of 100 data has an average (arithmetic mean) of 6 and a standard deviation of d, where d is positive. Which of the following pairs of data, when added to the list, must result in a list of 102 data with standard deviation less than d?
A. -6 and 0
B. 0 and 0
C. 0 and 6
C. 0 and 12
D. 6 and 6

"Standard deviation shows how much variation there is from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values."

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

According to the above adding two numbers which are closest to the mean will shrink the set most, thus decreasing SD by the greatest amount.

Closest to the mean are 6 and 6 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD.

Answer: E.

For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice:
https://gmatclub.com/forum/ps-questions- ... 85897.html
https://gmatclub.com/forum/ds-questions- ... 85896.html

Hope it helps.