Mann-Whitney U Table

What Is a Mann-Whitney U Table?

The Mann-Whitney U table provides critical values for the Mann-Whitney U test (also called the Wilcoxon rank-sum test). This non-parametric test compares two independent groups when the assumptions of the t-test are violated—particularly when data are ordinal or non-normally distributed.

When to Use the Mann-Whitney U Test

Choose this test when:

• Comparing two independent groups
• Data are ordinal or continuous but non-normal
• Sample sizes are small (n < 30)
• Outliers are present
• You want to test for differences in distribution or median

It's the non-parametric alternative to the independent samples t-test.

How the Test Works

1. Combine all observations from both groups
2. Rank them from smallest (1) to largest
3. Sum the ranks for each group (R₁, R₂)
4. Calculate U using:

$U_1 = n_1 n_2 + \frac{n_1(n_1+1)}{2} - R_1$

$U_2 = n_1 n_2 - U_1$

$U = \min(U_1, U_2)$

How to Read the Table

1. Let n₁ = smaller sample size, n₂ = larger sample size
2. Find the intersection of n₁ (row) and n₂ (column)
3. This gives the critical U value

Decision: If your calculated U ≤ critical U, reject H₀.

Example: Group A (n₁ = 8), Group B (n₂ = 12), α = 0.05 (two-tailed):

• Critical U = 22
• If calculated U = 18, reject H₀ (groups differ)

Worked Example

Data:

• Group A: 12, 15, 18, 20 (n₁ = 4)
• Group B: 22, 25, 28, 30, 35 (n₂ = 5)

Steps:

1. Combined ranks: 1, 2, 3, 4, 5, 6, 7, 8, 9

2. Group A ranks: 1, 2, 3, 4 → R₁ = 10

3. Group B ranks: 5, 6, 7, 8, 9 → R₂ = 35

4. Calculate U:

   • U₁ = (4)(5) + 4(5)/2 - 10 = 20 + 10 - 10 = 20
   • U₂ = (4)(5) - 20 = 0
   • U = min(20, 0) = 0

5. From table at α = 0.05: Critical U = 1

6. Since 0 ≤ 1, reject H₀

Handling Ties

When observations have the same value:

1. Assign them the average of the ranks they would occupy
2. Continue ranking subsequent values normally
3. Consider using a tie correction for the normal approximation

Normal Approximation (Large Samples)

For n₁ or n₂ > 20, use:

$\mu_U = \frac{n_1 n_2}{2}$

$\sigma_U = \sqrt{\frac{n_1 n_2 (n_1 + n_2 + 1)}{12}}$

$z = \frac{U - \mu_U}{\sigma_U}$

Compare z to standard normal critical values (±1.96 for α = 0.05, two-tailed).

Interpretation

The Mann-Whitney U test answers: "If I randomly pick one observation from each group, what's the probability that the observation from Group 1 exceeds the one from Group 2?"

The relationship is:

$P(\text{Group 1} > \text{Group 2}) = \frac{U}{n_1 n_2}$

Effect Size

The rank-biserial correlation measures effect size:

$r = 1 - \frac{2U}{n_1 n_2}$

Comparison with t-Test

Related Tools

• Mann-Whitney U Test Calculator — Run the test
• Two-Sample t-Test Calculator — Parametric alternative
• Wilcoxon Signed-Rank Test — For paired data
• Kruskal-Wallis Test — 3+ groups

Summary

The Mann-Whitney U table enables hypothesis testing for two independent groups without assuming normality. By comparing your calculated U statistic to the critical value, you can determine whether two groups come from different distributions. For larger samples, the normal approximation eliminates the need for tables, but for small samples (n ≤ 20), the exact critical values from this table ensure accurate Type I error control.