Nonparametric Tests

Mann-Whitney U Test Table: Critical Values for Two-Tailed α = 0.01

Exact critical values for the Mann-Whitney U test (Wilcoxon rank-sum test), the nonparametric alternative to the two-sample t-test.

  • n₁, n₂ = 2 to 20
  • Exact values
  • Exact p-value calculator
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Mann-Whitney U, α = 0.01

Rows: n₁ · Columns: n₂ · Cells: largest U that is significant at two-tailed α = 0.01. Reject H₀ if U ≤ the cell.

Mann-Whitney U Test Table (Critical Values, α = 0.01)
Size of the second sample (n₂) →
n₁ \ n₂ ↓		234567891011121314151617181920
2—————————————————00
3———————000111222233
4————001122334556678
5———01123456778910111213
6——0123456791011121315161718
7——013467910121315161819212224
8——1246791113151718202224262830
9—013579111316182022242729313336
10—0246911131618212426293134373942
11—02571013161821242730333639424548
12—13691215182124273134374144475154
13—137101317202427313438424549535760
14—147111518222630343842465054586367
15—258121620242933374246515560646973
16—259131822273136414550556065707479
17—2610151924293439444954606570758186
18—2611162126313742475358647075818792
1903712172228333945515763697481879399
20038131824303642485460677379869299105

Tip: click any value to highlight its row and column.

How to read this table

  1. Rank all n₁ + n₂ observations together. Let R₁ be the sum of the ranks in sample 1.
  2. Compute U₁ = R₁ − n₁(n₁ + 1)/2 and U₂ = n₁n₂ − U₁. Use U, the smaller of the two.
  3. Find the cell for your two sample sizes (the table is symmetric, so the order does not matter).
  4. Reject H₀ if U is less than or equal to the table value. A dash (—) means no result can be significant.

Worked example

A trial compares pain scores in 8 patients on a new treatment with 10 on standard care, and U = 14. Row n₁ = 8, column n₂ = 10 gives 11. Because 14 is larger than 11, the difference is not significant at two-tailed α = 0.01.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

U₁ = R₁ − n₁(n₁ + 1)/2  ·  U = min(U₁, n₁n₂ − U₁)  ·  reject H₀ if U ≤ Ucrit

Critical values come from the exact null distribution of U, obtained by enumerating all orderings of the two samples. Each value is the largest U with P(U ≤ u) ≤ α/2. The table assumes no ties.

Frequently asked questions

Is this the same as the Wilcoxon rank-sum test?

Yes. The Wilcoxon rank-sum statistic W = R₁ and the Mann-Whitney U are equivalent: U₁ = W − n₁(n₁ + 1)/2, so they always give the same p-value.

What about larger samples?

The calculator above gives exact results for samples up to 50 each. For large samples, U is approximately normal with mean n₁n₂/2 and variance n₁n₂(n₁ + n₂ + 1)/12.

What does the test actually compare?

It tests whether one group tends to have larger values than the other. It is only a test of medians when the two distributions have the same shape.

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