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: $f (q; r, v) = \frac{2 Γ (\frac{v + 1}{2})}{\sqrt{π} Γ (\frac{v}{2})} r q^{v - 1} \int_{0}^{\infty} t^{v} e^{- t^{2}} [Φ (\frac{q}{2} + \frac{t}{\sqrt{2}}) - Φ (\frac{t}{\sqrt{2}} - \frac{q}{2})]^{r - 2} d t .$

* Enter $q$ (q ≥ 0), number of groups $r$ (integer ≥ 2), and degrees of freedom $v$ (v > 0).

Step 1: Enter Parameters

Studentized Range,

q

e.g., 3

Number of Groups,

r

e.g., 4

Degrees of Freedom,

v

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: $f (q; r, v) = \frac{2 Γ (\frac{v + 1}{2})}{\sqrt{π} Γ (\frac{v}{2})} r q^{v - 1} \int_{0}^{\infty} t^{v} e^{- t^{2}} [Φ (\frac{q}{2} + \frac{t}{\sqrt{2}}) - Φ (\frac{t}{\sqrt{2}} - \frac{q}{2})]^{r - 2} d t .$

The calculator uses Simpson’s rule to numerically approximate the integral.

Studentized Range Distribution PDF Calculator

Step 1: Enter Parameters

Calculated PDF

How It Works

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