# 3.5.3.3.13 Srangeinv

## Definition:

$q=srangeinv(p, v, ir)$ computes the deviate, $x_p$, associated with the lower tail probability 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: $p(q)=C\int_0^\infty x^{\nu -1}e^{-\nu x^2/2}\{\Phi (y)[\Phi (y)-\Phi (y-qx)]^{r-1}dy\}dx$

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

For a given probability $p_0$, the deviate $q_0$ is found as the solution to the equation

$P(q_0)=p_0$

## Parameters:

p (output, double)
the probability.
v (input,double)
the number of degrees of freedom for the experimental error $\nu$. $\nu$ ≥ 1.0
ir (input, int)
the number of groups,$r$ .$ir \geq 2$
q (output, double)
the Studentized range statistic,$q$. $q>0.0$