Mann-Whitney U Test Calculator
Calculate the Mann-Whitney U test to compare two independent samples without normality assumptions.
Shows U statistic, z-score, and two-tailed p-value.
Mann-Whitney U Test (Wilcoxon Rank-Sum Test)
The Mann-Whitney U test compares two independent groups to determine if one tends to have larger values than the other. It is a non-parametric alternative to the independent samples t-test — no normality assumption needed.
When to use it:
- Two independent groups of data
- Data is not normally distributed
- Ordinal data (ranked categories)
- Comparing medians of two groups
The U statistic:
U1 = n1n2 + n1(n1+1)/2 - R1 U2 = n1*n2 - U1
Where n1, n2 are group sizes and R1 is the sum of ranks assigned to Group 1.
Normal approximation (for n1 or n2 > 20):
mu_U = n1 * n2 / 2 sigma_U = sqrt( n1n2(n1+n2+1) / 12 ) z = (U - mu_U) / sigma_U
Interpretation:
| p-value | Interpretation |
|---|---|
| p < 0.001 | Very strong evidence groups differ |
| p < 0.01 | Strong evidence groups differ |
| p < 0.05 | Moderate evidence groups differ |
| p >= 0.05 | Insufficient evidence groups differ |
Null hypothesis: Both groups come from the same distribution. A significant result means one group tends to produce higher values than the other.
Effect size (rank-biserial correlation): r = 1 - (2U) / (n1 * n2) Values near +1 or -1 indicate a large effect; near 0 indicates no effect.
Tie correction: When tied ranks exist, the standard deviation is adjusted: sigma_U = sqrt( n1n2/12 * (N+1 - sum(t^3-t) / (N(N-1))) ) where N = n1+n2 and t = size of each tied rank group.
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