The GMAT doesn't expect you to know how to calculate standard deviation (SD), but it does expect you to know what's needed to calculate it.

SD is the average of the distances between the mean of a set and all the values in the set. In other words, if we ever have enough data to find all these distances, we will know standard deviation.

Note before going into the statements that the numbers are consecutive integers.

Statement 1
We have 21 consecutive integers; no matter where the set begins, the spacing between the integers is constant. In other words, even without knowing the values of the numbers, we know exactly how the numbers are spread relative to the mean (the mean here is the 11th number).
Thus statement 1 is SUFFICIENT

Statement 2
This gives us the middle of the list. Without knowing how many terms are in the list, we cannot get a good idea of the spread of the numbers. for instance, there could be just 3 values (19, 20, 21). In this case, the distance between each value and the mean would be (1, 0, 1). The more values there are, the more long distances there will be, and the greater the SD will be (since SD is the average of the distances between the mean and all the values).
Thus statement 2 is NOT SUFFICIENT

The answer is A