Statistical Tables · free & public domain

Wilcoxon Signed-Rank Critical Values

Critical values for the Wilcoxon signed-rank test, conservative construction, n to 60.

How to use this table

Rank the absolute differences from smallest to largest, sum the ranks belonging to positive differences and those belonging to negative differences, and take the smaller sum. Reject H₀ if it is less than or equal to the tabled value. Example: n = 20 at two-tailed α = 0.05 → 52.

The table

Wilcoxon Signed-Rank Critical Values
two-tail α	0.10	0.05	0.02	0.01
n / one-tail α	0.05	0.025	0.01	0.005
5	0	—	—	—
6	2	0	—	—
7	3	2	0	—
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37
21	67	58	49	42
22	75	65	55	48
23	83	73	62	54
24	91	81	69	61
25	100	89	76	68
26	110	98	84	75
27	119	107	92	83
28	130	116	101	91
29	140	126	110	100
30	151	137	120	109
31	163	147	130	118
32	175	159	140	128
33	187	170	151	138
34	200	182	162	148
35	213	195	173	159
36	227	208	185	171
37	241	221	198	182
38	256	235	211	194
39	271	249	224	207
40	286	264	238	220
41	302	279	252	233
42	319	294	266	247
43	336	310	281	261
44	353	327	296	276
45	371	343	312	291
46	389	361	328	307
47	407	378	345	322
48	426	396	362	339
49	446	415	379	355
50	466	434	397	373
51	486	453	416	390
52	507	473	434	408
53	529	494	454	427
54	550	514	473	445
55	573	536	493	465
56	595	557	514	484
57	618	579	535	504
58	642	602	556	525
59	666	625	578	546
60	690	648	600	567

Conservative construction: the true error rate never exceeds α. Many printed tables instead pick whichever value lands nearest α, which can push the real rate above the level you asked for. n runs to 60.

How these values were produced

Exact enumeration of the signed-rank distribution (subset sums of 1…n), using the conservative rule.

Every one of these 224 values was computed from the underlying distribution — none was transcribed from a printed table. Values of this kind are mathematical facts: they come out the same for anyone who computes them correctly.

A deliberate difference. This table uses the conservative construction — the critical value is the last one whose true tail probability stays at or below α, so the real error rate never exceeds the level you asked for. Several widely printed versions instead pick whichever value lands nearest α, which can push the true rate above the nominal level. At n = 5, α = 0.05 a Wilcoxon test built that way actually runs at 0.0625.

Need the test run rather than the critical value looked up? The Proof Workbench computes t, r, χ² and ANOVA in your browser with assumption checks and a re-verifiable receipt.

Use it freely

This table is dedicated to the public domain under CC0 1.0. No attribution required, no signup, no limits — copy it into a lecture, a textbook, or your own site. A link back is welcome but not asked for.

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