Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ² of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states:[1]

${\displaystyle \sigma ^{2}\leq {\frac {1}{4}}(M-m)^{2}.}$

This equality holds precisely when half of the probability is concentrated at each of the two bounds.

Sharma et al. have sharpened Popoviciu's inequality:[2]

${\displaystyle {\sigma ^{2}+\left({\frac {\text{Third central moment}}{2\sigma ^{2}}}\right)^{2}}\leq {\frac {1}{4}}(M-m)^{2}.}$

Popoviciu's inequality is weaker than the Bhatia–Davis inequality which states

${\displaystyle \sigma ^{2}\leq (M-\mu )(\mu -m)}$

where μ is the expectation of the random variable.

In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality[3] gives a lower bound to the variance of the sample mean:

${\displaystyle \sigma ^{2}\geq {\frac {(M-m)^{2}}{2n}}.}$