Proof of variance of $S_n^2$

Question

Let

$$ S_n^2= \frac{1}{n} \sum_{i=1}^n X_i - \overline{X} $$

I'm looking for a clear simple proof of the variance of this just for personal knowledge to derive the $MSE$ later on any input would be appreciated.

Answer 1

Ok, I am assuming you know that the MSE is the (biased) sample variance: $MSE=\frac{1}{n}\sum_{i=1}^n (estimate_i-actual_i)^2$ when you replace your "actual" by the sample mean and the "estiamte" by your sample data. Also, your notation is a little ambiguous, as its not clear if the sample average is inside or outside of the summation. Numerically, it doesn't matter, so I will address both:

1) If $\overline{X}$ is outside of the summation, then note that $\frac{1}{n} \sum X_i = \overline{X}$. Therefore, your formula always evaluates to zero, so it has $0$ variance.

2) If $\overline{X}$ is inside the summation, then you get $\frac{1}{n} \sum X_i -\frac{1}{n} n\overline{X} = \frac{1}{n} \sum X_i -\overline{X}$, which is just case (1) above, so again you get zero.

The answer to your question is that the expression has zero variance.