Wilcoxon Rank Sum Test or Mann-Whitney U Test

Wilcoxon Rank Sum Test is also called Wilcoxon’s two-sample test or Wilcoxon-Mann-Whitney (WMW) test or Mann-Whitney-Wilcoxon (MWW) test and it is used for the comparison of two groups of non-parametric data or two independent samples. It compares the null hypothesis to the two-sided research hypothesis for differences or similarities. It is used for equal sample sizes and is used to test the median of two populations. Mann-Whitney U test is also used to compare two population means that come from the same population. It is similar to the t-test for continuous variables but can be used for ordinal data.

The assumptions of the Wilcoxon Rank Sum Test are as follows:

  • The dependent variable should be measured on a continuous scale or an ordinal scale.
  • The independent variable should be two independent, categorical groups.
  • Observations should follow the same shape as bell-shaped and skewed left and they are not normally distributed.
  • Observations should be no relationship between the two groups or within each group (independent).

The null and two-sided research hypotheses for the non-parametric tests are stated as follows:

Null hypothesis: Ho: The two populations are equal.

Alternative hypothesis: H₁: The two populations are not equal.

For Mann Whitney U test, the test statistic is denoted by ‘U’ which is a minimum of U, and U₂ For small samples, use the direct method to find the U statistic and for large samples, use the formula or technology like SPSS to run the test.

Wilcoxon Rank Sum Test


  • R₁ = Sum of the ranks for group 1
  • R₂ = Sum of the ranks for group 2
  • n₁ = Size of group 1
  • n₂ = Size of group 2

For example, low and high scores are approximately evenly distributed in the two groups, supporting the null hypothesis (groups are equal). If ranks of 2, 4, 6, 8, and 10 are assigned to the numbers of episodes of shortness of breath reported in the placebo group and ranks of 1, 3, 5, 7, and 9 are assigned to the numbers of episodes of shortness of breath reported in the new drug group.

  • R₁ = Sum of the ranks for group 1
  • R₁ = 2+4+6+ 8+10 = 30
  • R₂ = Sum of the ranks for group 2
  • R₂ = 1+3+5+7+9= 25


Wilcoxon Rank Sum Test
Wilcoxon's two-sample test

When there is no difference between populations, then U = 10.

Thus, small values of U support the research hypothesis and larger values of U support the null hypothesis.

In every test, U₁ + U₂ is always equal to n₁ x n₂.

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