e-Statistics

F-Distribution

Suppose that the sample variances and are obtained respectively from Group 1 and Group 2 with the respective sample sizes n and m, 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 pair (df1,df2) of numerator degree df1 = n-1 of freedom and denominator degree df2 = m-1 of freedom.

The F-distribution has a pair

(df1, df2) = ( , )

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

Upper-tailed region

$X > F_{\alpha,df1,df2} =$

The critical value $F_{\alpha,df1,df2}$ can be found in F-distribution table. Also, the lower-tailed region $X < F_{1-\alpha,df1,df2}$ can be obtained from $F_{\alpha,df2,df1}$ by applying the formula $F_{1-\alpha,df1,df2} = \displaystyle\frac{1}{F_{\alpha,df2,df1}}$ .

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