Mann-Whitney U Test

Non-parametric test to compare two independent groups — no normality assumption required.

The Mann–Whitney U test is the non-parametric alternative to the independent two-sample t-test. It compares two independent groups by ranking all the values together, so it works on ordinal data and makes no normality assumption. Use it when the t-test's normality assumption fails or your outcome is ranked rather than measured. A significant result means one group tends to produce larger values than the other. Results are validated against SciPy to full precision.

Input Groups

When to use it instead of a t-test

Choose Mann–Whitney when your data is ordinal (ranks, Likert scores), clearly non-normal with a small sample, or has outliers that would distort a mean. With larger samples (n ≳ 30 per group) the t-test is robust and usually preferred for its power; the which-test wizard can help you decide.

What it actually tests

The test asks whether one group's values tend to rank higher than the other's (stochastic dominance). It is often described as comparing medians, but that interpretation is only exact when the two distributions have the same shape — otherwise it compares overall location.

Assumptions

Worked example

Group A satisfaction ratings 3,4,4,5,5 versus Group B 1,2,2,3,4: Group A consistently ranks higher: U = 22.5 and p ≈ 0.043 (normal approximation with a tie correction), so the groups differ at α = 0.05 — though with only five values per group the evidence is modest. Paste two groups above for U and the p-value.

Common mistakes

Guide: Mann-Whitney U Test

The Mann-Whitney U test (also called the Wilcoxon rank-sum test) is a non-parametric statistical test used to compare two independent groups. Instead of comparing means directly, it ranks all the values from both groups together and tests whether one group tends to have higher ranks than the other.

It is the non-parametric equivalent of the independent-samples t-test and can be used whenever the t-test's normality assumption is in doubt.

Choose Mann-Whitney when:

  • Your data is ordinal (e.g., Likert scales, ranking data)
  • The distribution is heavily skewed or has significant outliers
  • Sample sizes are small and normality cannot be verified
  • Data comes from a non-normal distribution (e.g., reaction times, income)

If your data is approximately normally distributed and sample sizes are reasonable (>30 per group), the t-test is slightly more powerful and preferred.

The U statistic counts the number of times a value from Group 1 exceeds a value from Group 2 across all possible pairs. If U is close to its maximum (n₁ × n₂), Group 1 tends to be higher. If U is close to 0, Group 2 tends to be higher. If U ≈ n₁ × n₂ / 2, the groups are similar.

The p-value is more informative than U alone — it tells you whether the difference is statistically significant.

The effect size r is calculated as Z / √N where Z is the standardised U statistic and N is the total sample size. It ranges from 0 to 1:

  • r < 0.1: Negligible effect
  • 0.1 ≤ r < 0.3: Small effect
  • 0.3 ≤ r < 0.5: Medium effect
  • r ≥ 0.5: Large effect

A statistically significant result with a small effect size may not be practically meaningful.

The Mann-Whitney U test assumes:

  • The two samples are independent (different subjects in each group)
  • The variable is at least ordinal (values can be ranked)
  • Both distributions have the same shape (if testing medians)

It does not require normality or equal variances, which is its main advantage over the t-test.

Related Tools

T-Test (parametric) Effect Size Calculator Descriptive Statistics ANOVA