Mann-Whitney U & Wilcoxon signed-rank tests

 When we are comparing two groups that have an ordinal dependent variable or a continuous variable that does not meet the assumptions of the t-test (specifically not approximately normally distributed), we can look to the Mann-Whitney U test (also known as the Wilcoxon rank sum test), considered to be, though not always, the nonparametric alternative to the t-test. For example, you could have surveyed student’s perceptions toward two separate teaching styles using a 5-point scale, or determine if males and females at a certain company have similar salaries.

 The Mann-Whitney U test tests the null (H₀) and alternative (H_A) hypotheses:

 H₀: The probability that a randomly drawn observation from one population will be greater than a randomly drawn observation from the second observation is 50%. Or, the two samples belong to the same population with the same median (M).

M₁ = M₂

 H_A: The two samples have different medians and thus do not come from the same population.

M₁ ≠ M₂

 Alternatively, if the dependent variable is ordinal or not approximately distributed and the two groups are paired the Mann-Whitney U test is no longer valid and is replaced with the Wilcoxon signed-rank test. Similarly, the Wilcoxon signed-rank test is considered, though not always, the nonparametric alternative to the paired t-test. Examples of when the Wilcoxon signed-rank test may be appropriate include if we compare student responses to a survey on a 5-point scale before and after an exam or if patient blood pressures are reduced after given a new pharmaceutical.

  The Wilcoxon ranked sign test tests similar null and alternative hypotheses to the Mann-Whitney U test:

 H₀: The median of the population differences between the two paired groups is 0.

M(pop₁ - pop₂) = 0

 H_A: The median of the population differences is not 0.

M(pop₁ - pop₂) ≠ 0

 Before committing to the Mann-Whitney U or the Wilcoxon signed-rank test it should determined if the following assumptions are valid:

• The dependent variable is ordinal or continuous and the independent variable has two groups
• The observations are independent and randomly sampled
• Both groups have symmetrical distributions