naginterfaces.library.stat.prob_studentized_range(q, v, ir)[source]

prob_studentized_range returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

For full information please refer to the NAG Library document for g01em


, the Studentized range statistic.


, the number of degrees of freedom for the experimental error.


, the number of groups.


The probability associated with the lower tail of the distribution of the Studentized range statistic.

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There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.


The externally Studentized range, , for a sample, , is defined as:

where is an independent estimate of the standard error of the ’s. The most common use of this statistic is in the testing of means from a balanced design. In this case for a set of group means, , the Studentized range statistic is defined to be the difference between the largest and smallest means, and , divided by the square root of the mean-square experimental error, , over the number of observations in each group, , i.e.,

The Studentized range statistic can be used as part of a multiple comparisons procedure such as the Newman–Keuls procedure or Duncan’s multiple range test (see Montgomery (1984) and Winer (1970)).

For a Studentized range statistic the probability integral, , for degrees of freedom and groups can be written as:


The above two-dimensional integral is evaluated using quad.dim2_fin with the upper and lower limits computed to give stated accuracy (see Accuracy).

If the degrees of freedom are greater than the probability integral can be approximated by its asymptotic form:

This integral is evaluated using quad.dim1_inf.


NIST Digital Library of Mathematical Functions

Lund, R E and Lund, J R, 1983, Algorithm AS 190: probabilities and upper quartiles for the studentized range, Appl. Statist. (32(2)), 204–210

Montgomery, D C, 1984, Design and Analysis of Experiments, Wiley

Winer, B J, 1970, Statistical Principles in Experimental Design, McGraw–Hill