Studentized range distribution

Studentized range distribution

Parameters	k > 1 — the number of groups ν > 0 — degrees of freedom
Support	q ∈ [0; +∞)
PDF	${\displaystyle {\begin{matrix}f(q;k,\nu )={\frac {{\sqrt {2\pi }}k(k-1)\nu ^{\nu /2}}{\Gamma \left({\frac {\nu }{2}}\right)2^{\nu /2-1}}}\int _{0}^{\infty }x^{\nu }\varphi ({\sqrt {\nu }}x)\times \\[0.5em]\left[\int _{-\infty }^{\infty }\varphi (u)\varphi (u-qx)(\Phi (u)-\Phi (u-qx))^{k-2}\,{\text{d}}u\right]\,{\text{d}}x\end{matrix}}}$
CDF	${\displaystyle {\begin{matrix}F(q;k,\nu )={\frac {k\nu ^{\nu /2}}{\Gamma \left({\frac {\nu }{2}}\right)2^{{\frac {\nu }{2}}-1}}}\int _{0}^{\infty }x^{\nu -1}e^{-\nu x^{2}/2}\times \\[0.5em]\left[\int _{-\infty }^{\infty }\varphi (u)(\Phi (u)-\Phi (u-qx))^{k-1}\,{\text{d}}u\right]\,{\text{d}}x\end{matrix}}}$

In probability and statistics, Studentized range distribution is a continuous probability distribution that arises when estimating the range of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ) 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 random variable has a Studentized range distribution.

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