Sampling Distribution of the Variance of a Normally Distributed Random Sample

Topics: Statistics - Variance - Sampling Distribution

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(theorem)

Let $X_1,\dots ,X_n$ be iid random variables (random sample) with a normal distribution, expected value $\mu$ and variance ${\sigma }^2$.

Then:

$$\frac{n{S}_n^2}{{\sigma }^2}\sim {\chi }_n^2$$

…where:

$$S_n^2=\frac{{\sum }_{i=1}^n(x_i-\mu {)}^2}{n}$$

That is, $S_n^2$ is the sample variance.

Unknown Expected Value

When we don’t know the expected value of our random variables, do something similar (albeit with a slightly different sample variance).