# Conduct and Interpret a Mann-Whitney U-Test

What is the Mann-Whitney U-Test?

The Mann-Whitney U is a non-parametric test used to assess for significant differences in a scale or ordinal dependent variable by a single dichotomous independent variable.  It is the non-parametric equivalent of the independent samples t-test.  This means that the test does not assume any properties regarding the distribution of the dependent variable in the analysis.  This makes the Mann-Whitney U-test the appropriate analysis to use when analyzing dependent variables on an ordinal scale.  The Mann-Whitney U-test is also the mathematical basis for the H-test (also called Kruskal Wallis H).

The test was initially designed in 1945 by Wilcoxon for two samples of the same size and was further developed in 1947 by Mann and Whitney to cover different sample sizes.  Thus, the test is also called Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, Wilcoxon–Mann–Whitney test, or Wilcoxon two-sample test.

The U-test is a non-parametric test, in contrast to the t-test; it does not compare mean scores but median scores of two samples.  Thus, it is much more robust against outliers and heavy tail distributions.  Because the Mann-Whitney U-test is a non-parametric test, it does not require a special distribution of the dependent variable in the analysis.  Therefore, it is an appropriate test to compare groups when the dependent variable is not normally distributed and at least of ordinal scale.

For the test of significance of the Mann-Whitney U-test, it is assumed that with a large sample size, the distribution of the U-value approximates a normal distribution.  The U-value calculated with the sample can be compared against the normal distribution to calculate the confidence level.  The U-value represents the number of times observations in one sample precede observations in the other sample in ranking.

The Mann-Whitney U-Test in SPSS

An example research question for our U-Test is as follows:

Do the students that passed the exam achieve a higher grade on the standardized reading test?

The question indicates that the independent variable is whether the students have passed the final exam or failed the final exam, and the dependent variable is the grade achieved on the standardized reading test (A to F).

The Mann-Whitney U-Test can be found in Analyze/Nonparametric Tests/Legacy Dialogs/2 Independent Samples…

In the dialog box for the non-parametric two independent samples test, we select the ordinal dependent variable 'mid-term exam 1', and our nominal grouping variable 'Exam'.  Before the analysis can run, we need to specify the valid values for the grouping variable Exam, which in this case are 0 = fail and 1 = pass.  This can be done my clicking on 'Define Groups…'

We also need to select the Test Type.  The Mann-Whitney U-Test is marked by default.  Like the Mann-Whitney U-Test, the Kolmogorov-Smirnov Z-Test and the Wald-Wolfowitz runs-test have the null hypothesis that both samples are from the same population.  Moses extreme reactions test has a different null hypothesis: the range of both samples is the same.

If we select Mann-Whitney U, SPSS will calculate the U-value and Wilcoxon's W, which the sum of the ranks for the smaller sample.  If the values in the sample are not already ranked, SPSS will sort the observations according to the test variable and assign ranks to each observation.

The dialog box Exact… allows us to specify an exact non-parametric test of significance and the dialog box Options… defines how missing values are managed and if SPSS should output additional descriptive statistics.