The maximum variance of a finite dataset, given its mean, minimum, and maximum

Abstract

This paper derives the maximum variance of a finite dataset of real numbers, given their mean, minimum and maximum. An example is provided in which the maximum variance is less than half of the Bhatia-Davis upper bound, (maximum - mean)(mean - minimum). As the dataset length increases, the maximum variance under these constraints approaches this bound from below.

Added references. Changed "sequence" into "dataset". Rewrote part of the proof of Theorem 1 as Lemma 2. Improved the phrasing by using separate symbols for the variable (y) and the maximizer (x). Added examples. Emphasized that the bound is sharp. Added author information. Added MSC codes