View source for Mann-Whitney test

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A statistical test for testing the hypothesis  $  H _ {0} $
of homogeneity of two samples  $  X _ {1} \dots X _ {n} $
and  $  Y _ {1} \dots Y _ {m} $, 
all  $  m + n $
elements of which are mutually independent and have continuous distributions. This test, suggested by H.B. Mann and D.R. Whitney [[#References|[1]]], is based on the statistic

$$ 
U  =  W - 
\frac{1}{2}
 m ( m + 1 )  = \ 
\sum _ { i= } 1 ^ { n }  \ 
\sum _ { j= } 1 ^ { m } 
\delta _ {ij} ,
$$

where  $  W $
is the statistic of the [[Wilcoxon test|Wilcoxon test]] intended for testing the same hypothesis, equal to the sum of the ranks of the elements of the second sample among the pooled order statistics (cf. [[Order statistic|Order statistic]]), and

$$ 
\delta _ {ij}  = \ 
\left \{

Thus,  $  U $
counts the number of cases when the elements of the second sample exceed elements of the first sample. It follows from the definition of  $  U $
that if  $  H _ {0} $
is true, then

$$ \tag{* }
{\mathsf E} U  =  
\frac{nm}{2}
 ,\ \ 
{\mathsf D} U  =  
\frac{n m ( n + m + 1 ) }{12}
 ,
$$

and, in addition, this statistic has all the properties of the Wilcoxon statistic  $  W $, 
including asymptotic normality with parameters (*).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.B. Mann,   D.R. Whitney,   "On a test whether one of two random variables is statistically larger than the other"  ''Ann. Math. Stat.'' , '''18'''  (1947)  pp. 50–60</TD></TR></table>

====Comments====
Instead of Mann–Whitney test, the phrase  $  U $-
test is also used.