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Non-parametric Methods · MW-2S

Sample Size for Comparing Two Independent Groups Using the Mann–Whitney Test

Plan two independent groups using the probabilistic index and Noether’s large-sample Mann–Whitney approximation.

1. Information obtained from previous study / literature

Suggested convention: Group 1 is the control, reference, or usual-care group; Group 2 is the intervention, exposure, or new-treatment group. The probabilistic index is P(Group 2 > Group 1) + 0.5 P(tie).

Ordinal, discrete, rounded, or heavily heaped outcomes often have substantial ties.

Probabilistic-index interpretation aid
50%No probabilistic tendency; invalid for sample-size calculation
55%Small departure from the null
60%Group 2 is larger in 60% of pairwise comparisons
65%Group 2 is larger in 65% of pairwise comparisons

2. Information decided by the researcher

Tests whether the probabilistic index differs from 50%.

Enter 2 to recruit about twice as many Group 2 participants as Group 1 participants.

3. Results

Enter the hypothesis-test assumptions and select Calculate.

Formula, assumptions, limitations, and methodological note

Example clinical scenario

A clinical trial compares time to functional recovery between a new rehabilitation programme and usual care. Because recovery times are skewed, a Mann–Whitney comparison is planned.

Formula and methodological note

The calculator uses Noether’s large-sample approximation: N = (zα + zpower)² / [12 t(1−t)(p−0.5)²], where p is the probabilistic index and t is the planned Group 1 allocation fraction.

The Mann–Whitney test concerns relative stochastic tendency, represented here by the probabilistic index. It is not generally a test of equal medians. The Noether sample-size equation is a large-sample approximation for continuous outcomes with negligible ties. Substantial ties require a ties-aware method with additional planning information.

Highly imbalanced allocation should be feasible and scientifically justified. A probabilistic index close to 50% represents a small effect, so the required sample is sensitive to small changes in the assumed index.

Observations must be independent within and between two groups. This calculator does not cover paired observations, clustered designs, covariate adjustment, censored outcomes, exact finite-sample power, or non-inferiority/equivalence testing.

These tools support research planning and do not replace statistical consultation.

Method references

  1. Noether GE. Sample size determination for some common nonparametric tests. Journal of the American Statistical Association. 1987;82(398):645–647. doi:10.1080/01621459.1987.10478478.
  2. Happ M, Bathke AC, Brunner E. Optimal sample size planning for the Wilcoxon-Mann-Whitney test. Statistics in Medicine. 2019;38(3):363–375. doi:10.1002/sim.7983.
  3. Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods. 3rd ed. Wiley; 2014.
  4. Lehmann EL, D'Abrera HJM. Nonparametrics: Statistical Methods Based on Ranks. Springer; 2006.

Suggested citation

Yaacob, Najib Majdi. “Sample Size for Comparing Two Independent Groups Using the Mann–Whitney Test (MW-2S).” Web calculator. NajibMY, 2026. Accessed July 29, 2026. https://najibmy.netlify.app/tools/sample-size/comparing-two-independent-groups-mann-whitney/.