Editing Mann–Whitney U test (section)

===Proportion of concordance out of all pairs===
The following measures are equivalent.

====Common language effect size====
One method of reporting the effect size for the Mann–Whitney ''U'' test is with ''f'', the common language effect size.<ref name="Kerby2014">{{cite journal | last1 = Kerby | first1 = D.S. | year = 2014 | title = The simple difference formula: An approach to teaching nonparametric correlation | journal = Comprehensive Psychology | volume =  3| page =  11.IT.3.1| doi = 10.2466/11.IT.3.1 | s2cid = 120622013 | doi-access = free }}</ref><ref name="McGraw1992">{{cite journal | last1 = McGraw | first1 = K.O. | last2 = Wong | first2 = J.J. | year = 1992 | title = A common language effect size statistic | journal = Psychological Bulletin | volume = 111 | issue = 2| pages = 361–365 | doi = 10.1037/0033-2909.111.2.361 }}</ref> As a sample statistic, the common language effect size is computed by forming all possible pairs between the two groups, then finding the proportion of pairs that support a direction (say, that items from group 1 are larger than items from group 2).<ref name="McGraw1992"/> To illustrate, in a study with a sample of ten hares and ten tortoises, the total number of ordered pairs is ten times ten or 100 pairs of hares and tortoises. Suppose the results show that the hare ran faster than the tortoise in 90 of the 100 sample pairs; in that case, the sample common language effect size is 90%.<ref>{{Cite journal | author = Grissom RJ | year = 1994| title = Statistical analysis of ordinal categorical status after therapies | journal = [[Journal of Consulting and Clinical Psychology]] | volume = 62| issue = 2| pages = 281–284| doi= 10.1037/0022-006X.62.2.281 | pmid = 8201065}}</ref>

The relationship between ''f'' and the Mann–Whitney ''U'' (specifically <math>U_1</math>) is as follows:

:<math> f  = {U_1 \over n_1 n_2} \,</math>

This is the same as the [[#Area-under-curve (AUC) statistic for ROC curves|area under the curve (AUC) for the ROC curve]].

====''ρ'' statistic====
A statistic called ''ρ'' that is linearly related to ''U'' and widely used in studies of categorization ([[discrimination learning]] involving [[concept]]s), and elsewhere,<ref name="H1976" /> is calculated by dividing ''U'' by its maximum value for the given sample sizes, which is simply {{math|1=''n''<sub>1</sub>×''n''<sub>2</sub>}}. ''ρ'' is thus a non-parametric measure of the overlap between two distributions; it can take values between 0 and 1, and it estimates {{math|1=P(''Y'' > ''X'') + 0.5 P(''Y'' = ''X'')}}, where ''X'' and ''Y'' are randomly chosen observations from the two distributions. Both extreme values represent complete separation of the distributions, while a ''ρ'' of 0.5 represents complete overlap. The usefulness of the ''ρ'' statistic can be seen in the case of the odd example used above, where two distributions that were significantly different on a Mann–Whitney ''U'' test nonetheless had nearly identical medians: the ''ρ'' value in this case is approximately 0.723 in favour of the hares, correctly reflecting the fact that even though the median tortoise beat the median hare, the hares collectively did better than the tortoises collectively.{{citation needed|date=February 2012}}