Suppose the set $S$ contains $n$ positive integers. If the mean $\mu$ of the elements is known, is there a method to finding the maximum possible value of the standard deviation $\sigma$ for $S$?
I know there have to be a finite number of subsets of the positive integers that have $n$ elements and mean $\mu$; each element must be less than or equal to the sum of every number in the set, $n\mu$, so we have at most $(n\mu)^n$ possibilities for $S$. However, I can't think of a method that can reliably maximize the spread of $S$. For context, I'm specifically working with a set of 1000 positive integers with $\mu=10$, so brute forcing it is not an option. Any help is appreciated.