Recall that $s^2$, the sample variance, is the point estimate of $\sigma^2$, the population variance. When doing inference about the population variance, we are interested in the sampling distribution of $(n-1)s^2/\sigma^2$. The quantity $(n-1)s^2/\sigma^2$ has a chi-square distribution with $n-1$ degrees of freedom. Knowing the sampling distribution of this quantity, we can perform inference about the population standard deviatio, $s^2$. That is, we can create confidence intervals and conduct hypothesis tests about the population standard deviation.
| Chi-Square Distribution |
| $ \dfrac{(n-1)s^2}{\sigma^2} $ |
Like the t-distribution, the chi-square distribution depends on the degrees of freedom. Each different degrees of freedom corresponds to a different chi-square distribution. However, unlike the t-distribution, the chi-square isn't symmetric. While the t-distribution, as well as the normal distribution, are symmetric about their mean, the right tail of the chi-square distribution is longer than the left tail. This relative shape between the left and right side of the distribution are determined by the degrees of freedom.