Studentized Range Percent Point (Tukey HSD Critical Value) Calculator

Find the critical value q of the studentized range distribution (the Tukey HSD percent point) from probability p, number of groups k, and degrees of freedom, using numerical integration and bisection.

Input

Enter probability p, number of groups k, and degrees of freedom to find the critical value q of the studentized range distribution (the Tukey HSD percent point).

Probability p

A value strictly between 0 and 1 (usually 0.95)

Number of groups k

Number of groups being compared (integer of at least 2)

Degrees of freedom

Degrees of freedom of the error variance (at least 1)

Result

Critical value q for p = 0.95, k = 3 groups, df = 10

3.8767767

Groups k

Degrees of freedom

Lower probability P (Q at most q)

0.95

Upper probability

0.05

Probability density (approximate)

Cumulative distribution

How it works

The studentized range Q is the difference between the largest and smallest of k independent normal samples, divided by an independently estimated standard deviation. It supplies the critical value for the Tukey HSD test.
The critical value q satisfies P(Q at most q) equals p. This tool integrates the cumulative distribution with Gauss-Legendre quadrature and inverts it by bisection.
The cumulative distribution averages the infinite-degrees-of-freedom range distribution P(W at most q times s) over a scale s that follows the chi distribution with the given degrees of freedom.
Enter the number of groups k as an integer of at least 2, degrees of freedom of at least 1, and a probability p strictly between 0 and 1.
For very large degrees of freedom the distribution converges to the infinite case, and the computation switches to that approximation.

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