Sunday, 1 May 2016

statistics - Maximum standard deviation of n positive integers with set mean



Suppose the set S contains n positive integers. If the mean μ of the elements is known, is there a method to finding the maximum possible value of the standard deviation σ for S?




I know there have to be a finite number of subsets of the positive integers that have n elements and mean μ; each element must be less than or equal to the sum of every number in the set, nμ, so we have at most (nμ)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 μ=10, so brute forcing it is not an option. Any help is appreciated.


Answer



The most extreme case will be when n1 integers take the minimum value 1 and the remaining integer nμn+1. Any other case can have the standard deviation increased by moving a pair of values further apart



This has mean μ and standard deviation (using the population 1n method) of (μ1)n1



(If you insist on using the sample 1n1 method then the standard deviation would instead be (μ1)n)



In your example with n=1000 and μ=10, this would be 999 values of 1 and 1 value of 9001, with a standard deviation of about 284.423 (or 284.605)


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