Popoviciu's inequality on variances
Introduction In probability theory, '''Popoviciu's inequality''', named after Tiberiu Popoviciu, is an upper bound on the variance ''σ''2 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&ndash;Davis inequality]] Let  be a random variable with mean , variance , and  . Then, since , .Thus, .Now, applying the Inequality of arithmetic and geometric means, , with  and , yields the desired result:.
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