The RS_TEST function tests the hypothesis that two sample populations *X* and *Y* have the same median of distribution against the hypothesis that they differ. *X* and *Y* may be of different lengths. This type of test is often referred to as the “Wilcoxon Rank-Sum Test” or the “Mann-Whitney U-Test.”

The Mann-Whitney statistics for *X* and *Y* are defined as follows:

where *Nx* and *Ny* are the number of elements in *X* and *Y*, respectively, and *Wx* and *Wy* are the rank sums for *X* and *Y*, respectively. The test statistic Z, which closely follows a normal distribution for sample sizes exceeding 10 elements, is defined as follows:

This routine is written in the IDL language. Its source code can be found in the file rs_test.pro in the lib subdirectory of the IDL distribution.

## Examples

; Define two sample populations:

X = [-14, 3, 1, -16, -21, 7, -7, -13, -22, -17, -14, -8, $

7, -18, -13, -9, -22, -25, -24, -18, -13, -13, -18, -5]

Y = [-18, -9, -16, -14, -3, -9, -16, 10, -11, -3, -13, $

-21, -2, -11, -16, -12, -13, -6, -9, -7, -11, -9]

; Test the hypothesis that two sample populations, {xi, yi}, have

; the same median of distribution against the hypothesis in that

; they differ at the 0.05 significance level:

PRINT, RS_TEST(X, Y, UX = ux, UY = uy)

; Print the Mann-Whitney statistics:

PRINT, 'Mann-Whitney Statistics: Ux = ', ux, ', Uy = ', uy

IDL prints:

[1.45134, 0.0733429]

Mann-Whitney Statistics: Ux = 330.000, Uy = 198.000

The computed probability (0.0733429) is greater than the 0.05 significance level and therefore we do not reject the hypothesis that X and Y have the same median of distribution.

## Syntax

*Result* = RS_TEST( *X*, *Y* [, UX=*variable*] [, UY=*variable*] )

## Return Value

The result is a two-element vector containing the nearly-normal test statistic Z and the one-tailed probability of obtaining a value of the absolute value of Z or greater.

## Arguments

### X

An *n*-element integer, single-, or double-precision floating-point vector.

### Y

An *m*-element integer, single-, or double-precision floating-point vector.

## Keywords

### UX

Set this keyword to a named variable that will contain the Mann-Whitney statistic for *X*.

### UY

Set this keyword to a named variable that will contain the Mann-Whitney statistic for *Y*.

## Version History

4.0 |
Introduced |

## See Also

FV_TEST, KW_TEST, S_TEST, TM_TEST