Mathematical analysis derives the maximum variance for finite real sequences given mean and extreme values, showing convergence toward the Bhatia-Davis bound as sequence length increases.
This paper derives the maximum variance of a finite sequence 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 length of the sequence increases, the maximum variance under these constraints approaches this bound.
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Jules L. Ellis (2026) studied this question.
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