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

TL;DR

This paper derives the maximum possible variance for a finite dataset with known mean, minimum, and maximum, showing it approaches a specific bound as dataset size increases.

Contribution

It provides a closed-form derivation of the maximum variance under given constraints and compares it to the Bhatia-Davis bound, revealing asymptotic behavior.

Findings

Maximum variance is less than half of the Bhatia-Davis bound for small datasets.

Maximum variance approaches the Bhatia-Davis bound as dataset size increases.

The derived maximum variance formula improves understanding of data variability constraints.

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.