Popoviciu's inequality on variances

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In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu[citation needed], is an upper bound on the variance 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:[citation needed]

 \text{variance} \le \frac14 (M - m)^2.

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

Popoviciu's inequality is weaker than the Bhatia–Davis inequality.