Mann–Whitney Sample Size Calculator
Plan nonparametric experiments when data are skewed or ordinal. Enter the probability-of-superiority (Cliff’s delta converted), desired alpha and power, plus an optional allocation ratio to see how many participants are needed in each arm.
Approximation accuracy improves when both groups have at least 20 observations. Validate with simulation if your design violates large-sample assumptions.
Examples
- Probability-of-superiority 63.8%, α 5%, power 80%, equal allocation ⇒ Recruit 70 participants in the treatment arm and 70 in the control arm (total 140) to detect a 63.80% probability-of-superiority with 80.00% power at α = 5.00%.
- Probability-of-superiority 65%, α 5%, power 90%, equal allocation ⇒ Recruit 79 participants in the treatment arm and 79 in the control arm (total 158) to detect a 65.00% probability-of-superiority with 90.00% power at α = 5.00%.
FAQ
How do I convert Cohen’s d to probability-of-superiority?
Use the relation P = Φ(d/√2), where Φ is the standard normal CDF. For d = 0.5, P ≈ 0.638, matching the first example above.
Does this include continuity corrections?
No. For small samples you may add 1–2 participants per group as a safeguard or use exact power software to refine the estimate.
Can I set different alpha levels for one-sided tests?
Yes—enter the one-sided alpha directly. The calculator always uses a two-sided critical value internally, so halve α first if you need a one-sided design.
Additional Information
- Probability-of-superiority equals 0.5 + Cliff’s delta / 2. Convert from Cohen’s d via Φ(d/√2) if you prefer parametric effect sizes.
- The calculator uses Noether’s large-sample approximation for two-sided tests; increase the ceiling by a few participants when designing very small pilots.
- Allocation ratio > 1 means more participants in the treatment arm; use values < 1 to oversample the control group.