# Simple proof that variance of variance estimate is $\sigma^4 \cdot \frac{2}{n-1}$ given normal iid sample

Let $X_1,\dots X_n$ be i.i.d. and $N(\mu,\sigma^2)$ distributed. Let $$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$$ and $$S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i -\overline{X})^2$$

Then I know that through the distribution of $S^2$ one can very easily find its variance to be $$\mathbf{Var} (S^2)=\sigma^4 \cdot \frac{2}{n-1}$$

But given this result is everything i need, is it then possible to derive it using only elementary probability?

p.s. I didn't know what to tag the question, feel very free to help me out.

• math.stackexchange.com/questions/72975/… – Glen_b Apr 23 '14 at 9:45
• One answer to the question pointed out by @Glen_b (thanks!) applies even when the iid samples are from a non-normal distribution, and thus is even more beautiful and instructive. – Dilip Sarwate Apr 23 '14 at 12:44