The following theorem is important but its proof is long and difficult.
Sample variance from a normal distribution
If \(\{X_1, X_2, \dots, X_n\}\) is a random sample from a \(\NormalDistn(\mu, \sigma^2)\) distribution, the sample variance, \(S^2\) has a scaled Chi-squared distribution
\[ \frac {n-1}{\sigma^2} S^2 \;\sim\; \ChiSqrDistn(n - 1\;\text{df}) \](Proved in full version)
We will also write the distribution of \(S^2\) in the form
\[ S^2 \;\;\sim\;\; \frac{\sigma^2}{n-1} \;\times\; \ChiSqrDistn(n - 1\;\text{df}) \]Knowing the distribution of \(S^2\) lets you find probabilities relating to the variance or standard deviation of a sample from a normal distribution.
Example
In a random sample of \(n = 20\) values from a \(\NormalDistn(\mu,\;\sigma^2)\) distribution, what is the probability that the sample standard deviation will be more than 20% higher than the normal distribution's standard deviation, \(\sigma\)?
(Solved in full version)