e-Statistics

F-Distribution

The F distribution has a pair (df1 = , df2 = ) of numbers for the degree of freedom. The shape of the distribution is unimodal and skewed to the right, exhibiting a long right-hand tail. The critical point for the F distribution, denoted by $ F_{\alpha,df1,df2}$, corresponds to the upper tail region of level $ \alpha$.

The critical point $ F_{1-\alpha,df1,df2}$ can be found from $ F_{\alpha,df2,df1}$ via the formula $ F_{1-\alpha,df1,df2} = \displaystyle\frac{1}{F_{\alpha,df2,df1}}$. Thus, an lower-tailed region is related to the upper-tailed region.

Level (p-value) $ \alpha =$
Upper-tailed region $ X > F_{\alpha,df1,df2} =$

Suppose that the sample variances $ S_1^2$ and $ S_2^2$ are obtained respectively from Group 1 and Group 2 with the respective sample size $ n_1$ and $ n_2$, and that the two groups are independently observed and both satisfy the normality assumption. Then the statistic $ X = \displaystyle\frac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2}$ with true variances  $ \sigma_1^2$ and $ \sigma_2^2$ from the respective groups has the F distribution with degree $ (df1,df2) = (n_1-1, n_2-1)$ of freedom.