# 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]

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

If the sample size is finite then the von Szokefalvi Nagy inequality[3] gives a lower bound to the variance

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

where n is the sample size.

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.

A lower bound for the variance based on the Bhatia–Davis inequality has been found by Agarwal et al[4]

${\displaystyle (M-\mu )(\mu -m)-{\frac {(M-m)^{3}}{6}}\leq \sigma ^{2}}$

## References

1. ^ Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj). 9: 129–145.
2. ^ Sharma, R., Gupta, M., Kapoor, G. (2010). "Some better bound on variance with applications". Journal of Mathematical Inequalities. 4: 355–363. doi:10.7153/jmi-04-32.CS1 maint: Multiple names: authors list (link)
3. ^ Nagy JVS (1918) Uber algebraische Gleichungen mit lauter reellen Wurzeln, Jahresbericht der deutschen mathematiker-Vereingung, 27:37–43
4. ^ Agarwal RP, Barnett NS, Cerone P and Dragomir SS (2005) A survey on some inequalities for expectation and variance. Computers and mathematics with applications 49 (2005) 429-480