Chi-square Distribution

The chi-square distribution has the number of degrees of freedom (df) = . The curve lies on the positive line, and its shape is skewed to the right particularly when df is small. The critical point, denoted by $ \chi_{\alpha,df}$, is provided for the upper tail.

Level (p-value) $ \alpha =$
Upper-tailed region $ T > \chi_{\alpha,df} =$

When the sample variance $ S^2$ is obtained from the data of $ n$ observations which satisfies the normality assumption, the statistic $ X = \displaystyle\frac{(n-1)S^2}{\sigma^2}$ with true variance $ \sigma^2$ has the chi-square distribution with $ df = n-1$ degrees of freedom.

Conversely when the statistic $ X =$ is given, we can find the corresponding $ \alpha$ so that the value $ T$ belongs to the critical region, and call it p-value.