NAG Library Function Document

nag_deviates_studentized_range (g01fmc)


    1  Purpose
    7  Accuracy


nag_deviates_studentized_range (g01fmc) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.


#include <nag.h>
#include <nagg01.h>
double  nag_deviates_studentized_range (double p, double v, Integer ir, NagError *fail)


The externally Studentized range, q, for a sample, x1,x2,,xr, is defined as
q = maxxi - minxi σ^e ,  
where σ^e is an independent estimate of the standard error of the xi. 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, T-1,T-2,,T-r, the Studentized range statistic is defined to be the difference between the largest and smallest means, T-largest and T-smallest, divided by the square root of the mean-square experimental error, MSerror, over the number of observations in each group, n, i.e.,
q=T-largest-T-smallest MSerror/n .  
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, Pq;v,r, for v degrees of freedom and r groups, can be written as:
Pq;v,r=C0xv-1e-vx2/2 r-ϕyΦy-Φy-qx r-1dydx,  
C=vv/2Γ v/22v/2- 1 ,   ϕ y=12πe-y2/2   and   Φ y=-yϕ tdt.  
For a given probability p0, the deviate q0 is found as the solution to the equation
Pq0;v,r=p0, (1)
using a root-finding procedure. Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.


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


1:     p doubleInput
On entry: the lower tail probability for the Studentized range statistic, p0.
Constraint: 0.0<p<1.0.
2:     v doubleInput
On entry: v, the number of degrees of freedom.
Constraint: v1.0.
3:     ir IntegerInput
On entry: r, the number of groups.
Constraint: ir2.
4:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
The function was unable to find an upper bound for the value of q0. This will be caused by p0 being too close to 1.0.
On entry, ir=value.
Constraint: ir2.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, p=value.
Constraint: 0.0<p<1.0.
On entry, v=value.
Constraint: v1.0.


The returned solution, q*, to equation (1) is determined so that at least one of the following criteria apply.
(a) Pq*;v,r-p00.000005
(b) q0-q*0.000005×max1.0,q*.

Parallelism and Performance

nag_deviates_studentized_range (g01fmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for p1, where p1=p0r-1, so on input p should be set to p0r-1.


Three values of p, ν and r are read in and the Studentized range deviates or quantiles are computed and printed.

Program Text

Program Text (g01fmce.c)

Program Data

Program Data (g01fmce.d)

Program Results

Program Results (g01fmce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017