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 with a long right-hand tail. The critical point for the F distribution, denoted by $F_{\alpha,df1,df2}$ , is provided separately for the lower tail and the upper tail.

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

Level (p-value)	$\alpha =$
Lower-tailed region	$X < F_{1-\alpha,df1,df2} =$
Upper-tailed region	$X > F_{\alpha,df1,df2} =$

Suppose that the sample variances and are obtained respectively from Group 1 and Group 2 with the respective sample size and , 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 of freedom.