Studentized Range Distribution Inverse CDF Calculator
This calculator finds the quantile \( q \) such that the Studentized Range CDF satisfies: $$ F(q; r, v)=p. $$
* Enter a probability \( p \) (0 < \( p \) < 1), number of groups \( r \) (\( r \ge 2 \)), and degrees of freedom \( v \) (\( v > 0 \)).
Step 1: Enter Parameters
e.g., 0.5
e.g., 4
e.g., 20
How It Works
The CDF of the Studentized Range distribution is given by:
$$ F(q; r, v)=\int_0^q f(u; r, v)\,du, $$
where the PDF is: $$ f(q; r, v)=\frac{2\,\Gamma\Bigl(\frac{v+1}{2}\Bigr)}{\sqrt{\pi}\,\Gamma\Bigl(\frac{v}{2}\Bigr)}\,r\,q^{v-1}\int_{0}^{\infty}t^{v}e^{-t^2}\Bigl[\Phi\Bigl(\frac{q}{2}+\frac{t}{\sqrt{2}}\Bigr)-\Phi\Bigl(\frac{t}{\sqrt{2}}-\frac{q}{2}\Bigr)\Bigr]^{r-2}dt. $$
To find the quantile \( q \) such that \( F(q; r, v)=p \), the calculator uses a bisection method.
Simpson’s rule is used to numerically integrate the PDF.