Suppose the data for the first sample (Samp1) are 22, 24, 25, 29, 30 and the data for the second sample (Samp2) are 16, 21, 22, 23. The output from the Mann-Whitney test is:

N Median
C1 5 25.000
C2 4 21.500

Point estimate for η1 - η2 is 6.000
96.3 Percent CI for η1 - η2 is (0.002,13.002)
W = 33.5
Test of η1 = η2 vs η1 ≠ η2 is significant at 0.0500
The test is significant at 0.0491 (adjusted for ties)

The point estimate for η_{1} – η_{2} is the median of all possible pairwise differences between the two samples.

For this example, there are 5*4 = 20 pairwise differences. The possible pairwise differences for this example are: 22-16 = 6, 22-21 = 1, 22-22= 0, 22-23= -1, 8, 3, 2, 1, 9, 4, 3, 2, 13, 8, 7, 6, 14, 9, 8, 7.

You can get all the pairwise differences between 2 columns in Minitab by choosing

.The median of these differences is 6.

W = (number of positive differences) + 0.5(number of differences that equal 0) + 0.5(n_{1}(n_{1}+1)) where n_{1} = number of observations in the first sample.

For this example, W = 18 + 0.5(1) + 0.5*5*6 = 18 + 0.5 + 15 = 33.5.

The p-value is based on the test statistic for W. The test statistic, Z, (which is not part of the output) is a normal approximation using the mean and variance of W.

Mean of W = 0.5(n_{1} (n_{1} + n_{2} + 1)) variance of W = n_{1}*n_{2}(n_{1}+n_{2}+1)/12 where n_{1} and n_{2} are the number of observations in the first and second sample, respectively.

Z = (|W - mean of W| - .5)/square root of the variance of W.

Subtracting the .5 from the numerator is the continuity correction factor.

The p-value for H_{a}: η_{1} < η_{2} is CDF(Z). The p-value for H_{a}: η_{1} > η_{2} is (1 - CDF(Z)). The p-value for H_{a}: η_{1} ≠ η_{2} is 2*(1 - CDF(Z)). Where CDF is the cumulative probability of a standard normal distribution.

For this example:

- mean of W = 0.5*5(5+4+1) = 2.5*10 = 25
- variance of W = 5*4(5+4+1)/12 = 20*10/12 = 200/12 = 16.6667

Z = (|33.5 - 25| - .5)/sqrt(16.6667) = 1.9596

The p-value for H_{a}: η_{1} ≠ η_{2} is 2*(1 - 0.974979.) = 0.05.

You can get the cumulative probabilities in Minitab by choosing

.