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: 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: If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds 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 gives a lower bound to the variance of the sample mean:

Proof via the Bhatia–Davis inequality

Let A be a random variable with mean \mu, variance \sigma^2, and. Then, since , . Thus, . Now, applying the Inequality of arithmetic and geometric means,, with a = M - \mu and b = \mu - m, yields the desired result: .