does calculating the arithmetic mean of a serie of standar deviation measures make sense?

I'm trying to test if a hash function mantains a even distribution of values. My plan is to generate a set X of hashes and a integer Y symbolizing "slots" to see how many of the hashes maps to the slots given the mapping function x mod Y for every x in X. This is similar as how a hash table data structure works.

If every slot has a number y of mapped hashes, then I plan to calculate the standard deviation of these values respecting the expected value count(X) / Y. If repeat this test N times, I will end with N standar deviation values. Can I calculate then an arithmetic (maybe geometric?) mean to obtain a final value of how "evenly" distributed these hash function can be?

Adding the variances of your measurements to get an average variance would make more sense, this is because if $A$ and $B$ are random variables then Var(A+B) = Var(A)+Var(B) but the same is not true for standard deviations StDev(A+B) ≠ StDev(A)+StDev(B).