Studentized Range Distribution CDF Calculator
This calculator computes the cumulative probability for a Studentized Range distribution.
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 the Studentized Range value \( q \) (q ≥ 0), number of groups \( r \) (integer, \( r \ge 2 \)), and degrees of freedom \( v \) (v > 0).
Step 1: Enter Parameters
e.g., 3
e.g., 4
e.g., 20
How It Works
The PDF of the Studentized Range distribution is defined as:
$$ f(q; r, v)=\frac{2\,\Gamma\Bigl(\frac{v+1}{2}\Bigr)}{\sqrt{\pi}\,\Gamma\Bigl(\frac{v}{2}\Bigr)}\,r\,q^{v-1}\int_{0}^{\infty}t^{v}e^{-t^2}\Bigl[\Phi\Bigl(\frac{q}{2}+\frac{t}{\sqrt{2}}\Bigr)-\Phi\Bigl(\frac{t}{\sqrt{2}}-\frac{q}{2}\Bigr)\Bigr]^{r-2}dt. $$
To compute the CDF, we numerically integrate the PDF from 0 to \( q \):
$$ F(q; r, v)=\int_0^q f(u; r, v) \, du. $$
Simpson’s rule is used for numerical integration.