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:
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 the sample size is finite then the von Szokefalvi Nagy inequality gives a lower bound to the variance
where n is the sample size.
Popoviciu's inequality is weaker than the Bhatia-Davis inequality which states
where μ is the expectation of the random variable.
