Variance of sample variance?

What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.

I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$

I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as far as I got.

Here’s a general derivation that does not assume normality.

Let’s rewrite the sample variance $S^2$ as an average over all pairs of indices:

Since $\mathbb{E}[(X_i-X_j)^2/2]=\sigma^2$, we see that $S^2$ is an unbiased estimator for $\sigma^2$.

The variance of $S^2$ is the expected value of

When you expand the outer square, there are 3 types of cross product terms

depending on the size of the intersection $\{i,j\}\cap\{k,\ell\}$.

1. When this intersection is empty, the factors are independent and the expected cross product is zero.

2. There are $n(n-1)(n-2)$ terms where $|\{i,j\}\cap\{k,\ell\}|=1$ and each has an expected cross product of $(\mu_4-\sigma^4)/4$.

3. There are ${n\choose 2}$ terms where $|\{i,j\}\cap\{k,\ell\}|=2$ and each has an expected cross product of $(\mu_4+\sigma^4)/2$.

Putting it all together shows that Here $\mu_4=\mathbb{E}[(X-\mu)^4]$ is the fourth central moment of $X$.