# Mann Whitney Value Adjusted for Ties Minitab

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```					                                                          Technical Support Document
How Minitab Calculates the P-Value Adjusted for
Ties when Performing a Mann-Whitney Test
In the Mann-Whitney test, whenever ties occur in the combined data set, Minitab provides the p-value adjusted
for ties, in addition to the exact p-value. Suppose you are analyzing the following data set (from
EXH_STAT.mtw) using the Mann-Whitney test:

DBP1    DBP2
90      62
72      85
61      78
66      66
81      80
69      91
59      69
70      77

First, Minitab combines and ranks all the data:

Data Sample Rank
59    1     1.0
61    1     2.0
62    2     3.0
66    1     4.5
66    2     4.5
69    1     6.5
69    2     6.5
70    1     8.0
72    1     9.0
77    2     10.0
78    2     11.0
80    2     12.0
81    1     13.0
84    2     14.0
85    2     15.0
90    1     16.0
91    2     17.0

The combined data set has 8 + 9 = 17 values. However, not all 17 values are unique. The value 66 occurs twice,
and so does the value 69. A value that occurs more than once in the combined data set is a tie. This combined
data set has two ties.

When it encounters a tie, Minitab assigns the average rank to each observation in the tie. In this data set, 66
would be ranked in the 4th and 5th positions, so Minitab assigns the value an average rank of 4.5. Similarly, the
two observations of 69, which would fall into the 6th and 7th positions, are assigned an average rank of 6.5.

Technical Support Document
How Minitab Calculates the P-Value Adjusted for
Ties when Performing a Mann-Whitney Test
Next, Minitab calculates the sum of the ranks of the first sample, called W. In this example, W = 60.0.
(The formulas for calculating W can be found in Knowledgebase ID 1294. The Mann-Whitney statistic uses
pairwise differences. Because the Mann-Whitney statistic is a linear function of the Wilcox rank-sum statistic,
the two statistics are equivalent.)

After calculating W, Minitab calculates ZW, the standard normal approximation of W. If the combined data set
has ties, Minitab adjusts the denominator to calculate ZW. You can find the formulas for ZW and the adjusted
denominator in Minitab Help: Choose Stat > Nonparametric > Mann-Whitney, click on Help. Choose see
also > Methods and formulas.

The adjusted p-values are interpreted the same as the other p-values: If the p-value is less than alpha, then reject
the null hypothesis; if the p-value is greater than alpha, fail to reject the null hypothesis. The null hypothesis is
that both medians are equal. The alternative hypothesis is set by the user in the main dialog box of the Mann-
Whitney test.

The unadjusted p-value is conservative if ties are present; the adjusted p-value is usually more accurate, but it is
not always conservative.

Note: For details on calculating the point estimate for ETA1 – ETA2, W, and the unadjusted p-value, see
Knowledgebase ID 1294.