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.
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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).
A value strictly between 0 and 1 (usually 0.95)
Number of groups being compared (integer of at least 2)
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
3
Degrees of freedom
10
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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Studentized Range Percent Point (Tukey HSD Critical Value) Calculator