If set S consists of an odd number of even integers that have a normal

If that's what the book says, just throw it away. I don't think a single thing in that explanation is actually correct:

* even the question itself is flawed: finite sets can't have a 'normal distribution' (normal distributions are infinite distributions); at best they can have 'approximately normal' distributions. If that's the case, though, the set won't be equally spaced, as the explanation goes on to conclude. Further, normal distributions are simply never tested on the GMAT, so the question is irrelevant to test takers.

* just because your mean and median are equal is not enough to let you conclude that your set is equally spaced - that's nonsense. You can't even make the weaker conclusion that your set is 'symmetric' just from the fact that your mean and median are equal. For example, the set {-20,2,4,8,26} has a mean and median of 4, but is not equally spaced, nor is it even symmetric. I'd add that it is very useful to know, on many GMAT questions, that when you have an equally spaced set, your mean and median are always equal, but the converse of that fact is false - when your mean and median are equal, your set does not need to be equally spaced.

* The standard deviation of an equally spaced set is not straightforward to calculate. The standard deviation, for example, of the set {2, 4, 6} is certainly not equal to 2. To find the standard deviation, you find the distances from each element to the mean - those distances are 2, 0 and 2 - square them and average to get 8/3, then take the square root. So the standard deviation of {2, 4, 6} is just a bit less than \(\sqrt{2.7}\) and is certainly less than 2.

* The number of elements matters. The set {2, 4, 6} has a smaller standard deviation than the set {0, 2, 4, 6, 8}, because we can make this latter set by adding elements to the first set which are far from the mean.