Nonparametric Tests

Mann-Whitney U Test Table: Critical Values for Two-Tailed α = 0.05

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
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Mann-Whitney U, α = 0.05

Rows: n₁ · Columns: n₂ · Cells: largest U that is significant at two-tailed α = 0.05. Reject H₀ if U ≤ the cell.

Mann-Whitney U Test Table (Critical Values, α = 0.05)
Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

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

How to read this table

Rank all n₁ + n₂ observations together. Let R₁ be the sum of the ranks in sample 1.
Compute U₁ = R₁ − n₁(n₁ + 1)/2 and U₂ = n₁n₂ − U₁. Use U, the smaller of the two.
Find the cell for your two sample sizes (the table is symmetric, so the order does not matter).
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 17. Because 14 ≤ 17, the difference is significant at two-tailed α = 0.05.

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

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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 ≤ U_crit

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.

Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

Mann-Whitney U Test Table: Critical Values for Two-Tailed α = 0.05

Mann-Whitney U, α = 0.05

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

Related calculators and tutorials

Size of the second sample (n₂) → n₁ \ n₂ ↓	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	—	—	—	—	—	—	0	0	0	0	1	1	1	1	1	2	2	2	2
3	—	—	—	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	—	—	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	—	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	—	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	—	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	0	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	0	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	0	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	0	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	1	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	1	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	1	5	9	13	17	22	26	31	36	40	45	50	55	59	64	69	74	78	83
15	1	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	1	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	2	6	11	17	22	28	34	39	45	51	57	63	69	75	81	87	93	99	105
18	2	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	106	112
19	2	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	2	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127