# Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator.

I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get stuck finding the expected value of the $M_i\bar{M}$ term. Any clues?

I would also like to calculate the variance of the sample variance. In short I would like to calculate $\mathrm{Var}(M_i - \bar{M})^2$ but again that term rears its ugly head.

• Assuming the $M_i$ are iid, hint: let $\mu = E(M_1)$. Write $\sum (M_i - \bar M)^2 = \sum (M_i - \mu)^2 - n(\bar M - \mu)^2$ by throwing $\pm \mu$ into the difference and expanding. – guy Apr 3 '12 at 3:43
• See the first part of my answer here: math.stackexchange.com/questions/72975/… – user940 Apr 3 '12 at 4:01