Studentized range distribution

In probability and statistics, studentized range distribution is the continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population.

Studentized range distribution

Probability density function
Cumulative distribution function
Parameters	k > 1, the number of groups ${\displaystyle \nu }$ > 0, the degrees of freedom
Support	${\displaystyle q\in (0,+\infty )}$
PDF	${\displaystyle {\begin{matrix}f_{\text{R}}(q;k,\nu )={\frac {\,{\sqrt {2\pi \,}}\,k\,(k-1)\,\nu ^{\nu /2}\,}{\Gamma (\nu /2)\,2^{\left(\nu /2-1\right)}}}\int _{0}^{\infty }s^{\nu }\,\varphi ({\sqrt {\nu \,}}\,s)\,\times \\[0.5em]\left[\int _{-\infty }^{\infty }\varphi (z+q\,s)\,\varphi (z)\,\left[\Phi (z+q\,s)-\Phi (z)\right]^{k-2}\,\mathrm {d} z\right]\,\mathrm {d} s\end{matrix}}}$
CDF	${\displaystyle {\begin{matrix}F_{\text{R}}(q;k,\nu )={\frac {\,{\sqrt {2\pi \,}}\,k\,\nu ^{\nu /2}\,}{\,\Gamma (\nu /2)\,2^{\left(\nu /2-1\right)}}}\int _{0}^{\infty }s^{\nu -1}\,\varphi ({\sqrt {\nu \,}}\,s)\,\times \\[0.5em]\qquad \left[\int _{-\infty }^{\infty }\varphi (z)\,\left[\Phi (z+q\,s)-\Phi (z)\right]^{k-1}\,\mathrm {d} z\right]\,\mathrm {d} s\end{matrix}}}$

Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ²) and suppose that ${\displaystyle {\bar {y}}_{\min }}$ is the smallest of these sample means and ${\displaystyle {\bar {y}}_{\max }}$ is the largest of these sample means, and suppose s² is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution.

${\displaystyle q={\frac {{\overline {y}}_{\max }-{\overline {y}}_{\min }}{s/{\sqrt {n\,}}}}}$