Editing Mann–Whitney U test (section)

===Rank-biserial correlation===
A method of reporting the effect size for the Mann–Whitney ''U'' test is with a measure of [[rank correlation]] known as the rank-biserial correlation. Edward Cureton introduced and named the measure.<ref>{{cite journal | last1 = Cureton | first1 = E.E. | year = 1956 | title = Rank-biserial correlation | journal = Psychometrika | volume = 21 | issue = 3| pages = 287–290 | doi = 10.1007/BF02289138 | s2cid = 122500836 }}</ref> Like other correlational measures, the rank-biserial correlation can range from minus one to plus one, with a value of zero indicating no relationship.

There is a simple difference formula to compute the rank-biserial correlation from the common language effect size: the correlation is the difference between the proportion of pairs favorable to the hypothesis (''f'') minus its complement (i.e.: the proportion that is unfavorable (''u'')). This simple difference formula is just the difference of the common language effect size of each group, and is as follows:<ref name="Kerby2014"/>

:<math>r = f - u </math>

For example, consider the example where hares run faster than tortoises in 90 of 100 pairs. The common language effect size is 90%, so the rank-biserial correlation is 90% minus 10%, and the rank-biserial&nbsp;{{math|1=''r'' = 0.80}}.

An alternative formula for the rank-biserial can be used to calculate it from the Mann–Whitney ''U'' (either <math>U_1</math> or <math>U_2</math>) and the sample sizes of each group:<ref>{{cite journal | last1 = Wendt | first1 = H.W. | year = 1972 | title = Dealing with a common problem in social science: A simplified rank-biserial coefficient of correlation based on the ''U'' statistic | journal = European Journal of Social Psychology | volume = 2 | issue = 4| pages = 463–465 | doi = 10.1002/ejsp.2420020412 }}</ref>

: <math> r = f - (1 - f) = 2 f - 1 = {2U_1 \over n_1 n_2} - 1 = 1 - {2U_2 \over n_1 n_2} </math>

This formula is useful when the data are not available, but when there is a published report, because ''U'' and the sample sizes are routinely reported. Using the example above with 90 pairs that favor the hares and 10 pairs that favor the tortoise, ''U''<sub>2</sub> is the smaller of the two, so {{math|1=''U<sub>2</sub>'' = 10}}. This formula then gives {{math|1=''r'' = 1 – (2×10) / (10×10) = 0.80}}, which is the same result as with the simple difference formula above.