g01em returns the probability associated with the lower tail of the distribution of the Studentized range statistic.


public static double g01em(
	double q,
	double v,
	int ir,
	out int ifail
Visual Basic
Public Shared Function g01em ( _
	q As Double, _
	v As Double, _
	ir As Integer, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double g01em(
	double q, 
	double v, 
	int ir, 
	[OutAttribute] int% ifail
static member g01em : 
        q : float * 
        v : float * 
        ir : int * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: q, the Studentized range statistic.
Constraint: q>0.0.
Type: System..::..Double
On entry: v, the number of degrees of freedom for the experimental error.
Constraint: v1.0.
Type: System..::..Int32
On entry: r, the number of groups.
Constraint: ir2.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01em returns the probability associated with the lower tail of the distribution of the Studentized range statistic.


The externally Studentized range, q, for a sample, x1,x2,,xr, is defined as:
where σ^e is an independent estimate of the standard error of the xi'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, 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.,
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:
C=vv/2Γv/22v/2-1,  ϕy=12πe-y2/2  and  Φy=-yϕtdt.
The above two-dimensional integral is evaluated using d01da with the upper and lower limits computed to give stated accuracy (see [Accuracy]).
If the degrees of freedom v are greater than 2000 the probability integral can be approximated by its asymptotic form:
This integral is evaluated using d01am.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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

Error Indicators and Warnings

Note: g01em may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
On entry,q0.0,
There is some doubt as to whether full accuracy has been achieved.
An error occured, see message report.


The returned value will have absolute accuracy to at least four decimal places (usually five), unless ifail=2. When ifail=2 it is usual that the returned value will be a good estimate of the true value.

Parallelism and Performance


Further Comments



The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of q, ν and r.

Example program (C#): g01eme.cs

Example program data: g01eme.d

Example program results: g01eme.r

See Also