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Math Help - variance of the sample variance

  1. #1
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    variance of the sample variance

    Apologies if it is a trivial question, I googled but didn't find a direct answer. I'd like to check my guess

    I am given a random sample from a normal distribution with variance \sigma^2. I need to find variance of the sample variance, ie Var(S^2).

    I use the fact that, for a random sample from a normal population with variance \sigma^2,

    \frac{(n-1)S^2}{\sigma^2}\sim{\chi_{n-1}^2

    I found in the references that in the chi-squared distribution of n-1 degrees of freedom, the variance is 2(n-1)*

    Therefore

    Var(\frac{(n-1)S^2}{\sigma^2})=2(n-1)

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

    then find Var(S^2)=\frac{2{\sigma}^4}{n-1}

    Is that so?

    *By the way, is there a link somewhere to a simple derivation of the Variance (and the Mean) formula for the chi-squared distribution? My textbook at hand just quotes it without proof.
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  2. #2
    MHF Contributor matheagle's Avatar
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    V(S^2)=V\left(\left({(n-1)S^2\over \sigma^2}\right) \left( {\sigma^2\over n-1}\right)\right)

    NOW pull out the constant and square it....

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

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

    = {2\sigma^4\over n-1}
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  3. #3
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    that's clever! I suggest instead of (or in addition to) "Thanks" button to have the button "Like" - anyways I like your solution better than mine )))
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  4. #4
    MHF Contributor matheagle's Avatar
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    well if you do keep thanking me this much it may upset a certain bovine.
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