# Thread: What is the Varinace of the sample variance of iid normal distributed variables

1. ## What is the Varinace of the sample variance of iid normal distributed variables

Hi Guys
I have a sample of N iid normal distributed variables. And I need to calculate the Variance of the Maximumlikelihood estimated variance (sample variance) ${\sum (X_i-\bar X)^2\over N}$

Can somebody help me. I just can't come to an answer

Cheers
Todor

2. Hello,

This rv follows a $\chi^2(N-1)$ distribution So if you look at a table you should get its variance

3. Nope, the sum over $\sigma^2$ has a chi-square distribution.

${\sum_{i=1}^n(X_i-\bar X)^2\over n}= \left({\sigma^2\over n}\right)\left({\sum_{i=1}^n(X_i-\bar X)^2\over \sigma^2}\right)$

$\sim \left( {\sigma^2\over n}\right)\chi^2_{n-1}$

Hence $V\left( {\sum_{i=1}^n(X_i-\bar X)^2\over n}\right)= V\left( {\sigma^2\over n}\chi^2_{n-1}\right)$

$=\left( {\sigma^4\over n^2}\right)2(n-1)$

4. Originally Posted by matheagle
Nope, the sum over $\sigma^2$ has a chi-square distribution.
I wonder where I wrote that it doesn't