Editing Mann–Whitney U test (section)

==Normal approximation and tie correction==

For large samples, ''U'' is approximately [[normal distribution|normally distributed]]. In that case, the [[standard score|standardized value]]
:<math>z = \frac{ U - m_U }{ \sigma_U }, \, </math>

where ''m''<sub>''U''</sub> and ''σ''<sub>''U''</sub> are the mean and standard deviation of ''U'', is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution. ''m''<sub>''U''</sub> and ''σ''<sub>''U''</sub> are given by

:<math>m_U = \frac{n_1 n_2}{2}, \, </math> <ref name="auto">{{cite book |last1=Siegal |first1=Sidney (1956) |title=Nonparametric statistics for the behavioral sciences |publisher=McGraw-Hill |page=121}}</ref> and

:<math>\sigma_U=\sqrt{n_1 n_2 (n_1 + n_2+1) \over 12}. \, </math>  <ref name="auto"/> 

The formula for the standard deviation is more complicated in the presence of tied ranks. If there are ties in ranks, ''σ'' should be adjusted as follows:

:<math> \sigma_\text{ties}=\sqrt{ {n_1 n_2 (n_1 + n_2 +1) \over 12 }  - { n_1 n_2 \sum_{k=1}^K (t_k^3 - t_k) \over 12 n(n-1) } },\, </math> <ref>{{cite book |last1=Lehmann |first1=Erich | last2=D'Abrera | first2=Howard (1975) |title=Nonparametrics: Statistical Methods Based on Ranks |publisher=Holden-Day |page=20}}</ref>

where the left side is simply the variance and the right side is the adjustment for ties,  ''t''<sub>''k''</sub> is the number of ties for the ''k''th rank, and ''K'' is the total number of unique ranks with ties.

A more computationally-efficient form with {{math|1=''n''<sub>1</sub>''n''<sub>2</sub>/12}} factored out is

:<math> \sigma_\text{ties}=\sqrt{ {n_1 n_2 \over 12 } \left( (n+1) - { \sum_{k=1}^K (t_k^3 - t_k) \over n(n-1)} \right)},</math>

where {{math|1=''n'' = ''n''<sub>1</sub> + ''n''<sub>2</sub>}}.

If the number of ties is small (and especially if there are no large tie bands) ties can be ignored when doing calculations by hand. The computer statistical packages will use the correctly adjusted formula as a matter of routine.

Note that since {{math|1=''U''<sub>1</sub> + ''U''<sub>2</sub> = ''n''<sub>1</sub>''n''<sub>2</sub>}}, the mean {{math|1=''n''<sub>1</sub>''n''<sub>2</sub>/2}} used in the normal approximation is the mean of the two values of ''U''. Therefore, the absolute value of the ''z''-statistic calculated will be same whichever value of ''U'' is used.