Prove the sample variance is an unbiased estimator

Question

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 03 '12 at 03:43
See the first part of my answer here: http://math.stackexchange.com/questions/72975/variance-of-sample-variance/73080#73080 — , Apr 03 '12 at 04:01
Does this answer your question? Mean of $ \sum (X_i - \bar{X})^2$ — StubbornAtom, Jun 22 '23 at 06:29

score 2 · Answer 1 · answered Jun 04 '14 at 06:03

The only full and complete proof I could find on the internet can be found under http://economictheoryblog.wordpress.com/2012/06/28/latexlatexs2/

As goes over 70 (!) steps and shows every little formula manipulations I prefer rather not to post it here.

Prove the sample variance is an unbiased estimator

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