The  Wilcoxon Mann-Whitney is a 2 group non-parametric comparison test equivalent to the Parametric t-test that can be used to test treatment effects when data are not normally distributed.

• The Mann-Whitney U test, which may also be referred to as the Wilcoxon-Mann-Whitney test, or the Wilcoxon Rank-Sum test, evaluates the ranks of the combined scores from two independent groups.
• The Wilcoxon rank-sum test statistic (referred to as Ws if using the name Wilcoxon rank-sum) is based on using the sum of the ranks for observations drawn from one of the groups within the sample of data.

Generally, the groups being studied are designated as GROUP 1 = treatment group and GROUP 2 = control group. This statistic — regardless of whether you refer to it as the Mann-Whitney test or the Wilcoxon rank-sum test, is considered to be among the more powerful of the non-parametric statistical procedures; and when using large samples, the computational result of this test is generally the same as the parametric t-test for two independent groups.

When evaluating the outcome of this statistic, we can test the Wilcoxon rank-sum test statistic against a critical value from a table of standard values, or we can compute a z-score for the comparison of ranks.

In the Mann-Whitney U— Wilcoxon rank-sum test we compute a “z score” (and the corresponding probability of the “z score”) for the sum of the ranks within either the treatment or the control group.  The “U” value in this z formula is the sum of the ranks of the “group of interest” – typically the “treatment group”.

Essential Formulae

Below is the formula to compute z score for the Wilcoxon-Mann-Whitney test:

The formula to compute the probability of arriving at the z that you computed under the standard normal distribution (SND)  is shown in this next formula. We use this probability value to evaluate the outcome of the Mann-Whitney test. In this formula, replace the x term with z from the formula above. The value for π is 3.14, the value for σ² is 1, the value for μ is 0.

A working example:

You conducted a study to determine if a new treatment procedure was better than the standard method. 30 participants were recruited from a population of students and randomly allocated to either the new treatment procedure (T) or the standard method (S) so that the initial distribution was set at (n₁ = 15 and n₂ = 15).

After applying the treatments to each group respectively the students were ranked on a specific measure that demonstrates the influence of the two treatment methods. The ranks for each response score are given in the following table while maintaining the student’s group membership.

NT= New treatment; ST = Standard Treatment

Values in Row 2 in the table above represent the score on the dependent variable measuring the response to the two treatment types.

Values in Row 3 in the table above represent the codes for the group membership, where T=new treatment method group and S=standard method group.

Values in Row 3 in the table above represent the Rn= rank of participants within the total data set.

Ranks begin from the lowest score to the highest score.

What about ties?

In the case of a tie, we simply organize all of the data as in the table above, and then we assign each observation in a tie its average rank. So if we had two scores 12 and 12 and they had a rank of 3 and 4 then we would simply give the first value of 12 a rank of 3.5 and the second value of 12 a rank of 3.5.