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16
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3.4641
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The Mann-Whitney U Test Calculator is a non-parametric test for comparing two independent groups when the data do not meet the assumptions of the independent samples t-test. Also known as the Wilcoxon rank-sum test, it tests whether one group tends to have larger values than the other by comparing all possible pairs of observations between groups.
Enter values for two groups (up to 5 observations each) to compute the U statistic, its expected value under the null hypothesis, and a normal approximation Z-score for significance testing.
The Mann-Whitney U test counts how many times an observation from Group 1 exceeds an observation from Group 2 across all possible pairwise comparisons. For each pair, score 1 if the Group 1 value is larger, 0.5 for ties, and 0 otherwise:
$$U_1 = \sum_{i=1}^{n_1} \sum_{j=1}^{n_2} S(X_{1i}, X_{2j})$$
Where \(S(x, y) = 1\) if \(x > y\), \(0.5\) if \(x = y\), and \(0\) if \(x < y\). The complementary statistic is \(U_2 = n_1 n_2 - U_1\), and \(U_1 + U_2 = n_1 n_2\).
Under the null hypothesis of identical distributions, the expected value and standard deviation of U are:
$$\mu_U = \frac{n_1 n_2}{2}, \quad \sigma_U = \sqrt{\frac{n_1 n_2 (n_1 + n_2 + 1)}{12}}$$
For larger samples (both n > 10), the Z approximation works well:
$$Z = \frac{U_{\min} - \mu_U}{\sigma_U}$$
The test can also be computed using rank sums: \(U_1 = n_1 n_2 + n_1(n_1+1)/2 - R_1\), where \(R_1\) is the sum of ranks assigned to Group 1 in the combined sample.
Interpreting Mann-Whitney U test results:
Inputs
Results
Treatment group values tend to be higher. U₁ = 13, U₂ = 3. The Z approximation of -1.44 does not reach significance at α = 0.05 with these small samples.
Inputs
Results
Clear separation between groups. U<sub>min</sub> = 1 with Z = -2.61 < -1.96, indicating a significant difference at α = 0.05.
Use the Mann-Whitney U test when: (1) Data are ordinal rather than interval/ratio, (2) The normality assumption is violated and samples are small, (3) Data contain outliers that would disproportionately affect means, (4) The distribution shapes differ between groups. For large samples from approximately normal populations, the t-test has slightly more power than Mann-Whitney.
Strictly, Mann-Whitney tests whether the probability that a random observation from one group exceeds a random observation from the other equals 0.5 (P(X₁ > X₂) = 0.5). When both distributions have the same shape, this is equivalent to testing equal medians. When shapes differ, it tests stochastic dominance rather than location shift.
When observations from different groups are equal, each tied pair contributes 0.5 to the U count (instead of 0 or 1). For the rank-based calculation, tied values receive the average of the ranks they would occupy. A tie correction factor adjusts the variance formula for the Z approximation, though the effect is usually small unless ties are extensive.
They are equivalent tests — different formulations of the same procedure. The Wilcoxon rank-sum test uses the sum of ranks in one group (W = R₁), while Mann-Whitney uses the U count. They are related by: U₁ = W - n₁(n₁+1)/2. Both yield identical p-values and conclusions.
Yes. For a one-tailed test (e.g., H₁: Group 1 > Group 2), use U₁ directly. If U₁ is large (Group 1 values tend to exceed Group 2 values), compare the one-tailed Z to zα (e.g., 1.645 for α = 0.05). For the other direction, use U₂. The two-tailed test uses Umin and compares |Z| to zα/2.
Mann-Whitney can be applied with very small samples (as few as n₁ = n₂ = 3). However, with very small samples, only extreme results can achieve statistical significance. For n₁ = n₂ = 3, significance at α = 0.05 requires complete separation (U = 0). The Z approximation is reasonable when both n₁ and n₂ exceed 10.
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