# Variance of an unbiased estimator is 0 when the sample size goes to infinity

So I would like a proof for the following but I can't seem to do it myself.

I have a random variable $X$ and I draw $n$ samples($\{X_1, \ldots, X_n\}$) from it and I have

$$Z_n = \frac{\sum_{i = 1}^{n}(X_i - \overline{X_n})^2}{n-1}$$

so the random variable for the unbiased estimator for the variance. $\overline{X_n} = \frac{\sum_{i = 1}^{n}X_i}{n}$ is the variable for the sample average. I would like to prove the following:

$$\lim_{n \to \infty}Var(Z_n) = 0$$

I would like to prove this since it would mean that it worth to take large/larger samples since a variance of zero means that the random variable is equal to the mean with a probability of 1.

• This link has the proof you want. – Mur1lo Jun 27 '16 at 13:34
• Thanks. It only needs the weak law of large numbers is a prerequisite which is also simple to prove. – jakab922 Jun 27 '16 at 14:03