Inequalities giving lower bound on variance

Question

I'm interested in some inequalities giving bounds on variance, $\sigma^2$. For example Popoviciu's inequality says

If $m,M$ are the minimum and maximum values of a random variable $X$ with some underlying distribution, then $$\sigma^2 \leq \frac{(M-m)^2}{4}.$$

Similarly, the von Szokefalvi Nagy inequality gives a lower bound when the sample size, $n$ is finite. In that case:

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

Do you know any inequalities, analogous to this last one above, giving a lower bound on the variance, but without the condition of having a finite sample? Or alternatively, a reason why one should not expect such a convenient lower bound without the condition?

Answer 1

Yes!

Let $\mu$ be the arithmetic mean and $m_3$ the third central movement $m_3 \ = \ \sum_{i=1}^n({x_i - \mu})^3$ of a finite sample. Then: $$\sigma^2 \ + (\frac{m_3}{2\sigma^2})^2 \ \le \ \frac{(M-m)^2}{4}$$

Sources:

Sharma, R., Gupta, M., Kapoor, G. (2010). "Some better bounds on the variance with applications". Journal of Mathematical Inequalities