# 3.5.3.1.22 Srangecdf

## Definition:

$double prob = srangecdf(q, v, group)$ computes the probability associated with the lower tail of the distribution of the Studentized range statistic.

The externally Studentized range, $q$, for a sample, $x_1,x_2,\cdots,x_r$ is defined as:

$q=\frac{\max (x_i)-\min (x_i)}{\hat{\sigma _e}}$

where $\hat{\sigma _e}$ is an independent estimate of the standard error of the $x_i$ 's.

For a Studentized range statistic the probability integral,$P(q)$ , for $\nu$ degrees of freedom and $r$ groups, can be written as:

$P(q)=C\int_0^{+\infty }x^{\nu -1}e^{-\nu x^2/2}\{r\int_{-\infty }^{+\infty }\Phi (y)[\Phi (y)-\Phi (y-qx)]^{r-1}dy\}dx$

where $C=\frac{\nu ^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}$, $\Phi (y)=\int_{-\infty }^y\frac 1{\sqrt{2\pi }}e^{-t^2/2}dt$

## Parameters:

$q$ (input, double)
the Studentized range statistic,$q.$ $q>0$
$v$ (input, double)
the number of degrees of freedom for the experimental error. $\nu .$ $\nu \geq 1.0$
$group$ (input,int)
the number of groups,$r.group\geq 2$
$prob$(output, double)
the probability.