# Variance of the population variance.

• Nov 8th 2008, 10:50 AM
kman320
Variance of the population variance.
Xi's are iid with ~N(μ,σ^2). I am given the population variance.
http://farm4.static.flickr.com/3282/...6fc596de8b.jpg

How do you derive this (the variance part)?
http://farm4.static.flickr.com/3148/...86f10ea1d1.jpg
Thanks.
• Nov 10th 2008, 02:55 AM
mr fantastic
Quote:

Originally Posted by kman320
Xi's are iid with ~N(μ,σ^2). I am given the population variance.
http://farm4.static.flickr.com/3282/...6fc596de8b.jpg

How do you derive this (the variance part)?
http://farm4.static.flickr.com/3148/...86f10ea1d1.jpg
Thanks.

The very easy way is to know that the random variable $\displaystyle \frac{(n-1)S^2}{\sigma^2}$ has a $\displaystyle \chi^2$ distribution with (n-1) degrees of freedom. Therefore:

$\displaystyle Var \left( \frac{(n-1)S^2}{\sigma^2} \right) = 2(n-1)$

$\displaystyle \Rightarrow \left(\frac{(n-1)}{\sigma^2}\right)^2 Var (S^2) = 2(n-1)$

$\displaystyle \Rightarrow \frac{(n-1)^2}{\sigma^4} \, Var (S^2)= 2(n-1)$

and the result is obvious.

This also provides a very easy way of proving $\displaystyle E(S^2) = \sigma^2$.
• Nov 10th 2008, 09:07 AM
kman320
Thanks so much Mr. Fantastic. I was doing it the long and difficult way of Var(S^2)= E(S^4)-σ^4. The E(S^4) was taking forever.