mann whitney wilcoxon test

3 min read 18-03-2025

The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to compare two independent groups. Unlike parametric tests like the t-test, it doesn't assume that the data is normally distributed. This makes it a powerful tool when dealing with data that violates the assumptions of parametric tests. This guide will explore the test's purpose, assumptions, how to perform it, and its interpretation.

When to Use the Mann-Whitney U Test

The Mann-Whitney U test is the go-to choice when you want to compare the medians of two independent groups and:

Your data is not normally distributed: If your data significantly deviates from a normal distribution (as assessed by visual inspection of histograms or normality tests like Shapiro-Wilk), a non-parametric test like the Mann-Whitney U test is more appropriate.
Your data is ordinal: This test can handle ordinal data, where the values have a meaningful order but the intervals between them aren't necessarily equal. Examples include Likert scales (strongly agree to strongly disagree).
Your data contains outliers: Outliers can disproportionately influence parametric tests. The Mann-Whitney U test is less sensitive to outliers.
Your sample sizes are small: While it works well with larger samples, it's also suitable for smaller sample sizes where parametric tests might lack power.

Assumptions of the Mann-Whitney U Test

While more flexible than parametric tests, the Mann-Whitney U test still relies on certain assumptions:

Independent samples: The observations in each group must be independent of each other. This means that the measurement of one subject shouldn't influence the measurement of another.
Ordinal or continuous data: The data should be at least ordinal (ranked). Continuous data is acceptable.

How to Perform the Mann-Whitney U Test

The calculation of the Mann-Whitney U statistic can be done manually but is more often performed using statistical software like R, SPSS, or Python (with libraries like SciPy). Here's a conceptual overview:

Rank the data: Combine the data from both groups and rank all observations from lowest to highest. Assign the same rank to tied values (average the ranks).
Calculate the sum of ranks: Calculate the sum of ranks for each group separately (R1 and R2).
Calculate the U statistic: The U statistic is calculated using the following formula:

U1 = n1n2 + n1(n1+1)/2 - R1

U2 = n1n2 + n2(n2+1)/2 - R2

Where:
- n1 and n2 are the sample sizes of the two groups.
- R1 and R2 are the sums of the ranks for each group.
The smaller of U1 and U2 is the U statistic.
Determine the p-value: The U statistic is then compared to a critical value from a Mann-Whitney U table or calculated using statistical software. The p-value represents the probability of observing the obtained results (or more extreme results) if there were no difference between the groups.

Interpreting the Results

P-value < alpha (typically 0.05): You reject the null hypothesis. This suggests there's a statistically significant difference in the medians between the two groups.
P-value ≥ alpha: You fail to reject the null hypothesis. This suggests there's not enough evidence to conclude a statistically significant difference between the groups.

Example Scenario: Comparing Treatment Groups

Let's imagine a study comparing the effectiveness of two different treatments (A and B) for reducing pain. Pain scores are measured on a scale of 1 to 10. Because pain scores might not be normally distributed, a Mann-Whitney U test would be appropriate. If the p-value is below 0.05, we'd conclude that one treatment is significantly more effective in reducing pain than the other.

Mann-Whitney U Test vs. t-test

The key difference lies in the assumptions: the t-test assumes normality and equal variances, while the Mann-Whitney U test does not. Choose the Mann-Whitney U test when your data violates these assumptions.

Conclusion

The Mann-Whitney U test is a valuable non-parametric tool for comparing two independent groups when the assumptions of parametric tests are not met. Its robustness and relative simplicity make it a widely used statistical method across various fields. Remember to always interpret the results in the context of your research question and the limitations of the test. Using statistical software significantly simplifies the process and ensures accurate calculations.