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Suppose we don't know anything about a distribution except that it is a Normal distribution with mean $ \mu $ and variance $ \sigma^2 $.

Can we argue, theoretically, the sample variance, $ S_n^2 $ will be close to the population variance, and thus can we say something about the ratio $ \frac {S_n^2}{\sigma^2} $?

If this holds no basis in theory -- can we at least argue this in practice (mostly)?

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Normal distribution is fully defined by its mean and variance, that is, if you know $\mu$ and $\sigma^2$ you know everything about the distribution.

The data points in the sample are however drawn randomly, so the sample variance will be close to $\sigma^2$ in probabilistic sense. In the case of a normal distribution $S_n^2$ follows a $\chi^2$-distribution with $n-1$ degrees of freedom.

I suggest looking for derivations of Student t-test, which usually cover the relation between real and measured means and variances.

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