Let D be a (finite) set of data points. E.g., D might contain people’s heights, their exam results, or the temperature of an object at various times. Further, let m be D’s mean, and let s be D’s standard deviation. Then there’s a sense in which s is ‘symmetric’ around m: for one thing, values in m-s are usually considered just as ‘normal’ or ‘standard’ as the values in m+s. For another thing, we can ‘rotate’ D around m without affecting s: If each d ∈ D is replaced with d+2(m-d), the resulting set will have the same standard deviation as the original set. (E.g., if m = 10 and 7 ∈ D, then when we ‘rotate’ D around m, 7 would be replaced with 7+2(10-7) = 13.) Thirdly, I’ve noticed the following (slightly more interesting) fact:
For each d ∈ D, we can find a number d', s.th. if d is replaced with d' in D (while the rest of D stays the same), then the resulting set D' will have the same standard deviation as the original set.
The number d' in question will usually be close to, but distinct from, d+2(m-d). The mean of the new set will usually be different from the mean of the original set. (I say ‘usually’ because if all elements of D are the same, then d = d' = m = d+2(m-d).) As one would expect, D' can be ‘rotated’ again without affecting the standard deviation, and each element of the rotated D' can again be replaced without affecting the standard deviation; and so forth.
Now, my question is: are there other ways in which the standard deviation could be said to be ‘symmetric’, and is its ‘symmetry’ ever relevant in practice, when doing statistics/data analysis. Also, is there any theoretical interest/value in exploring the ‘symmetry’ of the standard deviation? (I’m very new to statistics, so please forgive me if the question has an obvious / boring answer.)