I have a set of data points. I extracted the Standard Deviation for the whole set, and also extracted STDs for a partition of the points where each element of the partition has at least two points. I averaged the subsets' STDs.
Of-course one can easily find examples where the averaged STDs are lower, equal, or greater than the 'full' STD. (for example, STD(1,1,11,11) is greater than average(STD(1,1), STD(11,11), and lower than average(STD(1,11), STD(1,11))).
However, I am interested in the relation between these two numbers. Is their expected value equal for random data? is there an equilibrium point? when does / doesn't averaging STDs 'make sense'?