Statistical Tables · free & public domain

Sign Test Critical Values

Critical values for the sign test, built on the conservative rule so the true error rate never exceeds α.

How to use this table

Count how many differences are positive and how many negative, discarding ties, and take the smaller of the two counts. Reject H₀ if that count is less than or equal to the tabled value. Example: n = 20 at two-tailed α = 0.05 → 5, so 5 or fewer of the less-frequent sign is significant.

The table

Sign Test Critical Values
two-tail α	0.01	0.02	0.05	0.10
n / one-tail α	0.005	0.01	0.025	0.05
5	—	—	—	0
6	—	—	0	0
7	—	0	0	0
8	0	0	0	1
9	0	0	1	1
10	0	0	1	1
11	0	1	1	2
12	1	1	2	2
13	1	1	2	3
14	1	2	2	3
15	2	2	3	3
16	2	2	3	4
17	2	3	4	4
18	3	3	4	5
19	3	4	4	5
20	3	4	5	5
21	4	4	5	6
22	4	5	5	6
23	4	5	6	7
24	5	5	6	7
25	5	6	7	7
26	6	6	7	8
27	6	7	7	8
28	6	7	8	9
29	7	7	8	9
30	7	8	9	10
31	7	8	9	10
32	8	8	9	10
33	8	9	10	11
34	9	9	10	11
35	9	10	11	12
36	9	10	11	12
37	10	10	12	13
38	10	11	12	13
39	11	11	12	13
40	11	12	13	14
41	11	12	13	14
42	12	13	14	15
43	12	13	14	15
44	13	13	15	16
45	13	14	15	16
46	13	14	15	16
47	14	15	16	17
48	14	15	16	17
49	15	15	17	18
50	15	16	17	18
51	15	16	18	19
52	16	17	18	19
53	16	17	18	20
54	17	18	19	20
55	17	18	19	20
56	17	18	20	21
57	18	19	20	21
58	18	19	21	22
59	19	20	21	22
60	19	20	21	23

Conservative construction: the true error rate never exceeds α. n runs to 60. A dash means no rejection is possible at that level.

How these values were produced

Exact Binomial(n, ½) tail probabilities (SciPy 1.17), 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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