Mann-Whitney U Test Calculator
The Mann-Whitney U test (also called the Wilcoxon rank-sum test) compares two independent groups without assuming normality. Enter the group sizes and the rank-sum of Group 1 to get the U statistic, z-score, and two-tailed p-value using the large-sample normal approximation.
Significance level (α)
Fail to reject H₀ — no significant difference detected at this α level.
- 1
U₁ statistic
10 × 12 + 10 × (10+1) ÷ 2 − 120 = 55 - 2
U₂ statistic
10 × 12 − 55 = 65 - 3
Test statistic U
min(55, 65) = 55 - 4
Mean μ_U
10 × 12 ÷ 2 = 60 - 5
Std dev σ_U
√(10 × 12 × (10+12+1) ÷ 12) = 15.1658 - 6
z-score
(55 − 60) ÷ 15.1658 = -0.3297Two-tailed p-value = 2 × (1 − Φ(|z|)) via the standard normal CDF.
How does this calculator work?
The Mann-Whitney U test ranks all observations together and computes U = min(U₁, U₂). Under the null hypothesis U follows a known distribution (approximated by a normal for n > 8). The two-tailed p-value tells you the probability of seeing a U this extreme or more by chance; p < α means the groups are significantly different.
Formula
How this is calculated
The Mann-Whitney U test ranks all observations from both groups together (1 to n₁+n₂), then computes U₁ = n₁·n₂ + n₁(n₁+1)/2 − R₁, where R₁ is the sum of ranks for Group 1. U₂ = n₁·n₂ − U₁ follows automatically, and U = min(U₁, U₂) is used for the test. Conceptually, U counts the number of times a Group 1 observation outranks a Group 2 observation (or vice versa), making it a measure of stochastic dominance between the groups.
For large samples (n₁ > 8 and n₂ > 8) U is approximately normally distributed under H₀ (the two populations have equal distributions) with mean μ_U = n₁·n₂/2 and standard deviation σ_U = √(n₁·n₂(n₁+n₂+1)/12). The standardised z-score is passed through the standard normal CDF to produce the two-tailed p-value. If the data contain many ties, a tie-correction factor reduces σ_U slightly; this calculator uses the untied formula, which is conservative.
For small samples the normal approximation is less reliable; consult exact U-distribution tables instead. As with all hypothesis tests, statistical significance (small p-value) does not imply practical importance — always report a measure of effect size alongside the test result.
Frequently asked questions
Use the Mann-Whitney U test when the data are ordinal, when the normality assumption of the t-test is seriously violated (especially with small samples), or when you want a test that is robust to outliers. The t-test is more powerful when both groups are genuinely normally distributed.
Pool all n₁ + n₂ observations and rank them from 1 (smallest) to n₁ + n₂ (largest), averaging ranks for tied values. R₁ is the sum of the ranks that belong to Group 1. In software like R, wilcox.test() returns the W statistic, which equals U₁ here. In Python, scipy.stats.mannwhitneyu() returns U₁ directly.
No. The tie correction reduces σ_U and produces a slightly larger (more significant) z-score. When many values are tied the correction can matter; if ties are numerous, use statistical software that implements the full tie-corrected formula or an exact permutation test.
Also known as
TG we-Calculate Editorial Team. (2026). Mann-Whitney U Test Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/u-test-calculator
TG we-Calculate Editorial Team. "Mann-Whitney U Test Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/u-test-calculator.
TG we-Calculate Editorial Team, "Mann-Whitney U Test Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/u-test-calculator
@misc{wecalculate_u_test_calculator, title = {Mann-Whitney U Test Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/u-test-calculator}}, year = {2026}, note = {TG we-Calculate} }
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