Mann–Whitney U Test

The Mann–Whitney U Test is a non-parametric statistical test used to compare two independent groups to determine whether their distributions differ — typically in terms of central tendency (median).

It’s an alternative to the independent samples t-test, but unlike the t-test, it does not assume normality of data.

In simple terms:

“The Mann–Whitney test checks if one group tends to have higher or lower values than another, without assuming the data are normally distributed.”

When to Use It

• Two groups - Independent (not paired) samples
• Data type - Ordinal or continuous (but not necessarily normal)
• Goal - Compare central tendency (medians) between groups
• Not for paired data - For paired data, use the Wilcoxon signed-rank test instead

Example Question

“Do customers from Region A spend more on average than customers from Region B?”

If the spending data are not normally distributed (e.g., skewed or contain outliers), the Mann–Whitney U test is the better choice than a t-test.

How It Works

Instead of comparing means directly, the Mann–Whitney test:

1. Combines all data from both groups
2. Ranks the data from smallest to largest (1 = smallest)
3. Calculates the sum of ranks for each group
4. Computes a U statistic — the number of times observations in one group precede observations in the other
5. Uses that U value to test whether the groups’ rank distributions are significantly different

Because it uses ranks, it’s robust to non-normal distributions and outliers.

Hypotheses

• H₀ (Null Hypothesis) - The two groups come from identical distributions (no difference in medians)
• H₁ (Alternative Hypothesis) - The two groups come from different distributions (one tends to be higher/lower)

Test Statistic

For samples of size n₁​ and n₂:

Where R₁​ = sum of ranks for group 1.
The smaller of U₁​ and U₂ is used as the test statistic.

For large samples (n_₁, n_₂ > 20), U is approximately normally distributed, so we can convert it to a Z-score.

Example in Python

Let’s say you want to test whether two marketing campaigns lead to different spending amounts.

import numpy as npfrom scipy.stats import mannwhitneyu​# Example data (non-normal)campaign_A = np.array([45, 50, 52, 60, 61, 62, 65, 70, 72])campaign_B = np.array([40, 42, 48, 51, 55, 57, 59, 61, 63])​# Perform Mann–Whitney U Teststat, p = mannwhitneyu(campaign_A, campaign_B, alternative='two-sided')​print(f"Mann–Whitney U Statistic: {stat:.3f}")print(f"P-value: {p:.3f}")

Interpretation:

• p < 0.05 → reject H₀ → significant difference between groups
• p ≥ 0.05 → fail to reject H₀ → no significant difference

Advantages

• Doesn’t assume normality
• Can handle ordinal data (ranks, ratings, etc.)
• Robust to outliers
• Works with unequal sample sizes

Limitations

• Assumes distributions have the same shape (only medians differ)
• Less powerful than t-test when data are normal
• Doesn’t tell you by how much the groups differ — only that they do

Python code

import pingouin as pgimport numpy as npimport pandas as pd​# Example: Two groups with non-normal distributionsnp.random.seed(42)df = pd.DataFrame({    'Col1': np.random.exponential(scale=2, size=30),    'Col2': np.random.exponential(scale=3, size=30)})​# Run Mann–Whitney U testresults = pg.mwu(x=df['Col1'], y=df['Col2'], alternative='two-sided')print(results)

Output:

     U-val alternative    p-val       RBC      CLESMWU  335.0   two-sided  0.09049 -0.255556  0.372222