Example 3: Confidence Interval for sample variance

For independent and identically distributed (i.i.d) variables $X\sim N(\mu,\sigma^2)$, the sample variance $s^2$ is the sum of squared normal variates, and is therefore distributed as $s^2/\sigma^2\sim \chi^2_{n-1}$. The C.I. for sample variance is therefore determined by the $\alpha/2$ and $1-\alpha/2$ quantiles of the chi-squared distribution with $n-1$ degrees of freedom. Because the chi-squared distribution is positively skewed, the C.I. is asymmetric with a bigger interval between the upper limit and the sample estimate than between the sample estimate and the lower limit.