Mann-Whitney U test calculator

The SciPy documentation example

The example data come from the SciPy manual for scipy.stats.mannwhitneyu: the ages at which nine randomly sampled young adults, five males and four females, were diagnosed with type II diabetes. SciPy takes the example from Rosie Shier, "Statistics: 2.3 The Mann-Whitney U Test", Mathematics Learning Support Centre, 2004. The table shows what this page calculates and what the SciPy v1.18.0 manual prints. The two asymptotic p values agree to 12 decimal places. SciPy manual, scipy.stats.mannwhitneyu.

With n = 5 and n = 4 there are 126 ways to split 9 ranks into groups of 5 and 4. The ranks of the males are 3, 5, 7, 8 and 9. Their rank sum is 3 + 5 + 7 + 8 + 9 = 32, so U1 = 32 − 5·6/2 = 17. The two-sided exact p value counts the splits with U ≤ 3 or U ≥ 17: 14 of 126, which is 1/9.

Ties and what SciPy does

• Equal values inside or between the samples get the average (mid) rank. The U statistic uses these mid ranks.
• With ties present, this page does not calculate the exact p value. This page computes the exact p value only when no value occurs more than once in the two samples together and each sample has at most 49 values. With ties, it shows only the tie-corrected asymptotic p values and says so.
• SciPy does not do the same thing in all modes. The SciPy manual says that method="exact" makes "No correction" for ties, and that mannwhitneyu "will not raise errors or warnings if there are ties". SciPy method="auto" "chooses 'exact' when the size of one of the samples is less than or equal to 8 and there are no ties; chooses 'asymptotic' otherwise." This page uses the tie-corrected variance n1·n2/12 · ((n + 1) − Σ(t³ − t)/(n(n − 1))), where t is the size of each group of equal values.
• R-devel wilcox.test uses a different rule: "an exact p-value is computed if the samples contain less than 50 finite values". With ties, R-devel does "exact inference" with "the conditional/permutation distribution". This page does not do that permutation calculation. With ties, it shows only the asymptotic p values. R manual, wilcox.test.

Effect size, the rank-biserial correlation

Alongside the p value the page reports the rank-biserial correlation r = 1 − 2U2/(n1·n2), which runs from −1 (every value in B is larger) to +1 (every value in A is larger). For the SciPy example r = 1 − 2·3/20 = 0.7. A p value measures the evidence against the null hypothesis that the two samples come from the same distribution; the rank-biserial tells you how strongly one group dominates. Fay and Proschan (2010) discuss the stochastic ordering and the Mann-Whitney functional views of the WMW test in "Wilcoxon-Mann-Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules", Statistics Surveys 4:1–39. Fay and Proschan 2010.

The calculator prints the midranks of each sample with every result, so you can verify the rank assignment by hand before you trust the p values.

Another view, the probability of a larger value

Divide U1 by n1·n2 and you get an estimate of p = P(A > B) + 0.5·P(A = B): the probability that a random value from sample A is larger than a random value from sample B, with half of the ties counted. The statsmodels manual defines this p for rank_compare_2indep, and its result object has methods for confidence intervals. For the SciPy example, U1/(n1·n2) = 17/20 = 0.85. statsmodels manual, rank_compare_2indep.

Mann-Whitney U or the two-sample t test?

The SciPy manual says: "When the means of samples from the populations are normally distributed, consider scipy.stats.ttest_ind." statsmodels states that the Mann-Whitney U test assumes equal variance or equal distribution under the null hypothesis. The chi-square test is not an alternative here: it works on category counts, not on raw values. For paired values of the same subjects, the SciPy manual names scipy.stats.wilcoxon.

Limits and accuracy

• The page calculates the exact p value with dynamic programming over all splits of the ranks. It shows the exact p value only when each sample has at most 49 values and no value occurs more than once. The two-sided exact p value is 2 × P(U ≥ max(U1, U2)), with a maximum of 1.
• The asymptotic p values use the tie-corrected standard deviation and the normal distribution, with and without the 0.5 continuity correction. With the correction, z = (max(U1, U2) − n1·n2/2 − 0.5)/σ, and the page uses z = 0 when this value is below 0. The SciPy manual gives the same form: z = (U − mn − d)/se.
• Values from −1,000,000,000 to 1,000,000,000 are accepted, with up to 10,000 values per sample. The exact p value is limited to 49 values per sample because the exact counting method stores (n1 + 1)(n1·n2 + 1) counts, so its memory grows quickly with the sample sizes. The calculation runs in your browser and makes no network request.
• The page rounds the rank sums, the U values, the rank-biserial r and U1/(n1·n2) to 12 significant digits. It shows the p values without rounding.

Disclaimers

• Results are statistical estimates and not medical, business or financial advice.
• The reference example is the published SciPy documentation example, and it names its source. The "Load a ties example" button loads 8 made-up values that only show how the page handles ties. The page has no data of this site.