Minitab calculates the Mann-Whitney statistic as follows:
Minitab ranks the two combined samples. Minitab gives the smallest observation rank 1, the second smallest observation rank 2, and so on.
If two or more observations are tied, Minitab assigns the average rank to both observations.
Minitab sums the ranks of the first sample.
Ranking tied values
Tied values occur when two or more observations are equal. If your data has tied values, Minitab ranks the data as follows:
Sort the observations in ascending order.
Assign ranks to each observation as if there were no ties.
For a tied set, take the average of the corresponding ranks and assign this value as the new rank to each tied value in that set.
A sample has 9 observations: 2.4, 5.3, 2.4, 4.0, 1.2, 3.6, 4.0, 4.3, and 4.0.
(assuming no ties)
Minitab also uses the following information to calculate the test statistics:
The number of the set of ties is 2.
The number of tied values that is in the 1st set is 2.
The number of tied values that is in the 2nd set is 3.
The Mann-Whitney test uses a normal approximation method to determine the p-value of the test.
is approximately distributed as a normal distribution with a mean of 0 and a standard deviation of 1, N(0,1).
The normal approximation p-value for the three alternative hypotheses uses a continuity correction of 0.5.
H1: η1 > η2
H1: η1 < η2
H1: η1 ≠ η2
If your data contains ties, then, Minitab adjusts the p-value by replacing the denominator of the above Z statistics by:
The adjusted p-value is usually more accurate than the unadjusted p-value. However, the unadjusted p-value is the more conservative estimate because it is always greater than the adjusted p-value for a specific pair of samples.
Mann-Whitney test statistics
size of sample 1
size of sample 2
median of sample 1
median of sample 2
1, 2, …, I
number of sets of ties
number of tied values in the ith set of ties
The approximation algorithm that Minitab uses to calculate the point estimate of η1 – η2is described in this article: J.W. McKean and T.A. Ryan, Jr. (1977). "An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two Sample Location Problem", Transactions on Mathematical Software, 183–185.
The confidence interval of η1 – η2 is defined as the range of values of η1 – η2 for which the null hypothesis is not rejected.
The method that Minitab uses to calculate the confidence interval is described in this article: J.W. McKean and T.A. Ryan, Jr. (1977). "An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two Sample Location Problem", Transactions on Mathematical Software, pp.183-185.