Mann-Whitney U Test

Nonparametric test comparing two independent groups when data doesn't meet normality assumptions.

Test Overview & Methodology

The Mann-Whitney U test (also known as Wilcoxon rank-sum test) compares the distributions of two independent groups by ranking all observations and comparing rank sums. It's ideal when data is ordinal or when parametric assumptions are violated.

Advantages:

• • No normality assumption required
• • Robust to outliers and extreme values
• • Works with ordinal data effectively
• • High efficiency compared to t-test
• • Handles unequal sample sizes well

When to Use:

• • Two independent groups comparison
• • Non-normal distributions
• • Ordinal or continuous data
• • Small sample sizes
• • Presence of outliers

U Statistic & Rank-Sum Formulas

Test Methodology:

• • Step 1: Combine and rank all observations from both groups
• • Step 2: Calculate sum of ranks for each group ($R_1$, $R_2$)
• • Step 3: Calculate both U statistics
• • Step 4: Test statistic is the smaller U value, compared via exact tables or the normal approximation

Mann-Whitney U statistics

= sum of ranks assigned to group 1

= sum of ranks assigned to group 2

= sample size of group 1

= sample size of group 2

U1 + U2 = n1 · n2. Report the smaller of the two as the test statistic U.

Normal approximation (for n1, n2 > ~10, no large tie correction needed)

= expected value of U under the null hypothesis

= standard deviation of U under the null hypothesis

Key Assumptions

• • The two samples are independent random samples
• • Observations are at least ordinal (can be rank-ordered)
• • Under the null hypothesis, the two distributions have the same shape and spread (for a location-shift interpretation)
• • Ties are handled with mid-ranks and a corresponding variance correction

Reading the Mann-Whitney Rank Comparison Plot

Example Output & Result Interpretation

Two suppliers' delivery times (n₁ = 8, n₂ = 8) are ranked together. Group A's rank sum is R₁ = 52, giving U₁ = 52 − 8(9)/2 = 16 and U₂ = 64 − 16 = 48. The smaller value, U = 16, is compared against the critical value or converted to a z-score for a large-sample approximation.

p ≤ α — Reject H₀

• • Rank distributions differ significantly between groups
• • One group tends to have systematically higher/lower values
• • Report the effect size alongside the p-value

p > α — Fail to Reject H₀

• • No significant evidence that group distributions differ
• • Observed rank-sum difference is consistent with chance

Effect Size & Alternative Names:

• • Effect size (r): r = Z/√N
• • Small effect: r = 0.1
• • Medium effect: r = 0.3
• • Large effect: r = 0.5
• • Alternative names: Wilcoxon rank-sum, Two-sample Wilcoxon

sixsigmacenter.com Implementation Guide

Example Data Layout for Mann-Whitney U Test

Two independent groups for Mann-Whitney U test. No normality assumption required. Effective for ordinal data or when parametric assumptions are violated. Sample Size Recommendation: Minimum 5 observations per group, preferably 10+ for better approximation to normal distribution of test statistic.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.