Editing Mann–Whitney U test (section)

===Illustration of calculation methods===

Suppose that [[Aesop]] is dissatisfied with his [[The Tortoise and the Hare|classic experiment]] in which one [[tortoise]] was found to beat one [[hare]] in a race, and decides to carry out a significance test to discover whether the results could be extended to tortoises and hares in general. He collects a sample of 6 tortoises and 6 hares, and makes them all run his race at once. The order in which they reach the finishing post (their rank order, from first to last crossing the finish line) is as follows, writing T for a tortoise and H for a hare:
:T H H H H H T T T T T H
What is the value of ''U''?
* Using the direct method, we take each tortoise in turn, and count the number of hares it beats, getting 6, 1, 1, 1, 1, 1, which means that {{math|1=''U<sub>T</sub>'' = 11}}. Alternatively, we could take each hare in turn, and count the number of tortoises it beats. In this case, we get 5, 5, 5, 5, 5, 0, so {{math|1=''U<sub>H</sub>'' = 25}}. Note that the sum of these two values for {{math|1=''U'' = 36}}, which is {{math|6×6}}.
* Using the indirect method:
: rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth.
: the sum of the ranks achieved by the tortoises is {{math|1=12 + 6 + 5 + 4 + 3 + 2 = 32}}.
:: Therefore {{math|1=''U<sub>T</sub>'' = 32 − (6×7)/2 = 32 − 21 = 11}} (same as method one).
:: The sum of the ranks achieved by the hares is {{math|1=11 + 10 + 9 + 8 + 7 + 1 = 46}}, leading to {{math|1=''U<sub>H</sub>'' = 46 − 21 = 25}}.
<!--
===Illustration of object of test===
A second example race illustrates the point that the Mann–Whitney ''U'' test does not test for inequality of [[median]]s, but rather for difference of distributions. Consider another hare and tortoise race, with 19 participants of each species, in which the outcomes are as follows, from first to last past the finishing post:

:H H H H H H H H H T T T T T T T T T '''T''' '''H''' H H H H H H H H H T T T T T T T T T

If we simply compared medians, we would conclude that the median time for tortoises is less than the median time for hares, because the median tortoise here (in bold) comes in at position 19, and thus actually beats the median hare (in bold), which comes in at position 20. However, the value of ''U'' is 100 (using the quick method of calculation described above, we see that each of 10 tortoises beats each of 10 hares, so {{math|1=''U'' = 10×10}}). Consulting tables, or using the approximation below, we find that this ''U'' value gives significant evidence that hares tend to have lower completion times than tortoises ({{math|''p'' < 0.05}}, two-tailed). Obviously these are extreme distributions that would be spotted easily, but in larger samples something similar could happen without it being so apparent. Notice that the problem here is not that the two distributions of ranks have different [[variance]]s; they are mirror images of each other, so their variances are the same, but they have very different [[skewness]].
-->