Mann-Whitney U Test

mann-whitney

wilcoxon-rank-sum

non-parametric

ranks

Non-parametric comparison of two independent groups on an ordinal or non-normal continuous outcome

Published

April 17, 2026

Research question

The Mann-Whitney U test (equivalent to the Wilcoxon rank-sum test) asks whether two independent groups differ in distribution, using ranks instead of raw values. It is the default when an independent-samples comparison is needed but the outcome is ordinal (e.g., Likert-style pain rating) or strongly non-normal (e.g., cytokine concentrations with a long right tail). Biomedical examples: (1) are post-operative pain scores on days 1-7 different between open and laparoscopic cholecystectomy patients?; (2) do serum ferritin levels differ between patients with and without iron-deficiency anaemia?

Assumptions

Assumption	How to verify in R
Two independent groups	design
Outcome at least ordinal	scale level
Similar shape in both groups (for a median-shift interpretation)	overlaid density / boxplot

If the two distributions have very different shapes, the Mann-Whitney still tests stochastic dominance but cannot be interpreted as a difference in medians alone.

Hypotheses

\[H_0: P(X_1 > X_2) = P(X_2 > X_1) \qquad H_1: P(X_1 > X_2) \ne P(X_2 > X_1)\]

Under the shape-equality assumption, this reduces to \(H_0: \text{median}_1 = \text{median}_2\).

R code

library(tidyverse); library(rstatix); library(effectsize); library(ggstatsplot)
set.seed(42)

# Ferritin (ng/mL) in 28 controls and 22 iron-deficient patients
fer <- tibble(
  group    = factor(rep(c("Control", "ID"), c(28, 22)), levels = c("Control", "ID")),
  ferritin = c(rlnorm(28, log(80), 0.5),
               rlnorm(22, log(18), 0.6))
)

# Inspect shapes
fer |> group_by(group) |> get_summary_stats(ferritin, type = "five_number")

fer |> ggplot(aes(x = group, y = ferritin)) +
  geom_boxplot(fill = "#2A9D8F", outlier.colour = "#F4A261") +
  labs(y = "Serum ferritin (ng/mL)") + theme_minimal()

# Mann-Whitney U test
fer |> wilcox_test(ferritin ~ group, detailed = TRUE)

# Effect size: rank-biserial correlation r (-1 to 1)
fer |> wilcox_effsize(ferritin ~ group)
# or equivalently
effectsize::rank_biserial(ferritin ~ group, data = fer)

# Visualisation with inline stats
ggbetweenstats(data = fer, x = group, y = ferritin, type = "nonparametric",
               xlab = "Group", ylab = "Ferritin (ng/mL)")

Interpreting the output

With \(W\) = 534 and \(p < .001\), the null of equal distributions is rejected. The rank-biserial \(r \approx 0.77\) indicates a large effect. Inspection of the boxplot confirms the patient group’s ferritin distribution lies almost entirely below the control distribution.

Effect size

Rank-biserial correlation \(r_{rb}\) = \(2 \times U / (n_1 n_2) - 1\). Range \([-1, 1]\); Cohen’s thresholds (adapted): small 0.10, medium 0.30, large 0.50.

Reporting (APA 7)

Serum ferritin was significantly lower in the iron-deficient group than in controls (Mann-Whitney U = 534, p < .001, rank-biserial r = .77). Median ferritin was 17.5 ng/mL in iron-deficient patients vs. 81.2 ng/mL in controls.

Common pitfalls

Reporting mean and SD in a Mann-Whitney analysis invites inconsistency; report medians and IQRs.
With ties, R uses a normal approximation; an exact permutation test (coin::wilcox_test) is cleaner for small samples.
Treating unequal distribution shapes as “median differences” overstates what the test actually shows.

Parametric vs. non-parametric alternative

Parametric alternative: independent-samples t-test.
For paired data: Wilcoxon signed-rank test.
For three or more groups: Kruskal-Wallis test.