Studentized Range Distribution PDF Calculator
This calculator computes the PDF of the Studentized Range distribution used in multiple comparisons (e.g., Tukey’s HSD test).
The PDF is given by: $$ f(q; r, v)=\frac{2\,\Gamma\Bigl(\frac{v+1}{2}\Bigr)}{\sqrt{\pi}\,\Gamma\Bigl(\frac{v}{2}\Bigr)}\,r\,q^{v-1}\int_{0}^{\infty}t^{v}e^{-t^2}\Bigl[\Phi\Bigl(\frac{q}{2}+\frac{t}{\sqrt{2}}\Bigr)-\Phi\Bigl(\frac{t}{\sqrt{2}}-\frac{q}{2}\Bigr)\Bigr]^{r-2}dt. $$
* Enter \( q \) (q ≥ 0), number of groups \( r \) (integer ≥ 2), and degrees of freedom \( v \) (v > 0).
Step 1: Enter Parameters
e.g., 3
e.g., 4
e.g., 20
How It Works
The Studentized Range distribution is used to determine the probability that the maximum difference among a set of means exceeds a given value.
Its PDF is expressed as: $$ f(q; r, v)=\frac{2\,\Gamma\Bigl(\frac{v+1}{2}\Bigr)}{\sqrt{\pi}\,\Gamma\Bigl(\frac{v}{2}\Bigr)}\,r\,q^{v-1} \int_{0}^{\infty}t^{v}e^{-t^2}\Bigl[\Phi\Bigl(\frac{q}{2}+\frac{t}{\sqrt{2}}\Bigr)-\Phi\Bigl(\frac{t}{\sqrt{2}}-\frac{q}{2}\Bigr)\Bigr]^{r-2}dt. $$
The calculator uses Simpson’s rule to numerically approximate the integral.