F

The F cumulative distribution function is given by the integral

$\displaystyle \limfunc$FDist(x;n, m) = $\displaystyle {\frac{{\Gamma (\frac{n+m}{2})}}{{\Gamma (\frac{n}{2}%
)\Gamma (\frac{m}{2})}}}$$\displaystyle \left(\vphantom{ \frac{n}{m}}\right.$$\displaystyle {\frac{{n}}{{m}}}$$\displaystyle \left.\vphantom{ \frac{n}{m}}\right)^{{\frac{n}{2}}}_{}$$\displaystyle \int_{{0}}^{{x}}$u$\scriptstyle {\frac{{n-2}}{{2}}}$$\displaystyle \left(\vphantom{ 1+\frac{n}{m}u}\right.$1 + $\displaystyle {\frac{{n}}{{m}}}$u$\displaystyle \left.\vphantom{ 1+\frac{n}{m}u}\right)^{{-\frac{n+m}{2}}}_{}$du

of the probability density function

$\displaystyle \limfunc$FDen(u;n, m) = $\displaystyle {\frac{{\Gamma (\frac{n+m}{2})}}{{\Gamma (\frac{n}{2}%
)\Gamma (\frac{m}{2})}}}$$\displaystyle \left(\vphantom{ \frac{n}{m}}\right.$$\displaystyle {\frac{{n}}{{m}}}$$\displaystyle \left.\vphantom{ \frac{n}{m}}\right)^{{\frac{n}{2}}}_{}$u$\scriptstyle {\frac{{n-2}}{{2}}}$$\displaystyle \left(\vphantom{ 1+\frac{n}{m}u}\right.$1 + $\displaystyle {\frac{{n}}{{m}}}$u$\displaystyle \left.\vphantom{ 1+\frac{n}{m}u}\right)^{{-\frac{n+m}{2}}}_{}$

The variable x can be any positive number, and n and m can be any positive integers. The F distribution is used to determine the validity of the assumption of identical standard deviations of two normal populations. It is the distribution on which the analysis of variance procedure is based.

The inverse distribution function $\limfunc$FInv(p;n, m) gives the value of x for which the integral $\limfunc$FDist(x;n, m) has the value p. These function names automatically turn gray when they are entered in mathematics mode. The relationship between these two functions is illustrated in the following examples.


$\displaystyle \limfunc$FDist(.1;3, 5) =   4.3419×10-2  
$\displaystyle \limfunc$FInv(.043419;3, 5) =  .1  
$\displaystyle \limfunc$FDist(3.7797;2, 5) =  .9  
$\displaystyle \limfunc$FInv(.9;2, 5) =  3.7797  

Standard F distribution tables list some of the values of the inverse F distribution function. Thus, for example, the 4.4th percentile for the F distribution having degrees of freedom $\left(\vphantom{ 3,5}\right.$3, 5$\left.\vphantom{ 3,5}\right)$ is $\func$FInv(.044;3, 5) = .1, and the 90th percentile for the F distribution having degrees of freedom $\left(\vphantom{ 2,5}\right.$2, 5$\left.\vphantom{ 2,5}\right)$ is $\func$FInv(.90;2, 5) = 3.7797.

The following plots show distribution functions $\limfunc$FDist(x;n, m) and density functions $\func$FDen(x;n, m) for $\left(\vphantom{ n,m}\right.$n, m$\left.\vphantom{ n,m}\right)$ = $\left(\vphantom{ 1,1}\right.$1, 1$\left.\vphantom{ 1,1}\right)$,$\left(\vphantom{ 2,5}\right.$2, 5$\left.\vphantom{ 2,5}\right)$,$\left(\vphantom{ 3,15}\right.$3, 15$\left.\vphantom{ 3,15}\right)$, and 0≤x≤5.

itbpFU2.1577in1.7573in0inF density functions itbpFU2.1577in1.7573in0inF distribution functions