Editing Mann–Whitney U test (section)

===Comparison to Student's ''t''-test===
The Mann–Whitney ''U'' test tests a null hypothesis that the [[probability distribution]] of a randomly drawn observation from one group is the same as the probability distribution of a randomly drawn observation from the other group against an alternative that those distributions are not equal (see [[Mann–Whitney U test#Assumptions and formal statement of hypotheses]]). In contrast, a [[t-test]] tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney ''U'' test and the t-test do not test the same hypotheses and should be compared with this in mind.  
;Ordinal data: The Mann–Whitney ''U'' test is preferable to the ''t''-test when the data are [[Level of measurement#Ordinal scale|ordinal]] but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.
;Robustness:As it compares the sums of ranks,<ref name="Motulsky 2007">Motulsky, Harvey&nbsp;J.; ''Statistics Guide'', San Diego, CA: GraphPad Software, 2007, p. 123</ref> the Mann–Whitney ''U'' test is less likely than the ''t''-test to spuriously indicate significance because of the presence of [[outlier]]s. However, the Mann–Whitney ''U'' test may have worse [[Type I and type II errors|type I error]] control when data are both heteroscedastic and non-normal.<ref>{{Cite journal|last=Zimmerman|first=Donald W.|date=1998-01-01|title=Invalidation of Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions|journal=The Journal of Experimental Education|volume=67|issue=1|pages=55–68|doi=10.1080/00220979809598344|issn=0022-0973}}</ref>
;Efficiency:When normality holds, the Mann–Whitney ''U'' test has an (asymptotic) [[Efficiency (statistics)|efficiency]] of 3/{{pi}} or about&nbsp;0.95 when compared to the ''t''-test.<ref name="Lehmann 1999">Lehamnn, Erich&nbsp;L.; ''Elements of Large Sample Theory'', Springer, 1999, p. 176</ref> For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney ''U'' test is considerably more efficient than the ''t''.<ref name="Conover 1980">Conover, William&nbsp;J.; [https://onlinepubs.trb.org/onlinepubs/nchrp/cd-22/manual/v2chapter6.pdf ''Practical Nonparametric Statistics''], John Wiley & Sons, 1980 (2nd Edition), pp. 225–226</ref> This comparison in efficiency, however, should be interpreted with caution, as Mann–Whitney and the t-test do not test the same quantities. If, for example, a difference of group means is of primary interest, Mann–Whitney is not an appropriate test.<ref>{{Cite journal|last1=Lumley|first1=Thomas|last2=Diehr|first2=Paula|author2-link=Paula Diehr|last3=Emerson|first3=Scott|last4=Chen|first4=Lu|date=May 2002|title=The Importance of the Normality Assumption in Large Public Health Data Sets|journal=Annual Review of Public Health|volume=23|issue=1|pages=151–169|doi=10.1146/annurev.publhealth.23.100901.140546|pmid=11910059| doi-access=free|issn=0163-7525}}</ref>

The Mann–Whitney ''U'' test will give very similar results to performing an ordinary parametric two-sample [[t test|''t''-test]] on the rankings of the data.<ref>{{cite journal |doi=10.2307/2683975 |title=Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics |first1=William&nbsp;J. |last1=Conover |first2=Ronald&nbsp;L. |last2=Iman |author-link2=Ronald L. Iman |journal=[[The American Statistician]] |volume=35 |issue=3 |year=1981 |pages=124–129 |jstor=2683975 }}</ref>
{| class="wikitable float-right"
|+Relative efficiencies of the Mann–Whitney test versus the two-sample ''t''-test if ''f'' = ''g'' equals a number of distributions<ref>{{Cite book |last=Vaart |first=A. W. van der |url=http://dx.doi.org/10.1017/cbo9780511802256 |title=Asymptotic Statistics |date=1998-10-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511802256 |isbn=978-0-511-80225-6}}</ref>
!Distribution
!Efficiency
|-
|Logistic
|<math>\pi^2/9</math>
|-
|Normal
|<math>3/\pi
</math>
|-
|Laplace
|3/2
|-
|Uniform
|1
|}