Statistical Tables · free & public domain

Runs Test Critical Values

Lower and upper critical run counts at α = 0.05 for testing a sequence for randomness.

How to use this table

Count the runs — each unbroken stretch of the same symbol is one run. Find n₁ (count of one symbol) down the side and n₂ (the other) across the top. Reject randomness if the number of runs is ≤ the lower value or ≥ the upper value. Example: n₁ = 10, n₂ = 10 → reject if runs ≤ 6 or ≥ 16. Too few runs means clustering; too many means over-alternation.

The table

bound lower — reject if runs ≤
Runs Test Critical Values — lower — reject if runs ≤
n₁ / n₂2345678910111213141516171819202122232425
2——————————22222222222222
3————22222222333333333333
4———222333333334444444444
5——2233333444444455555555
6—22333344445555556666666
7—22333445555566666677777
8—23334455566666777778888
9—23344555666777788888899
10—233455566777788889999910
11—2344556677788899991010101010
122234456677788899910101010111111
13223455667788999101010101111111112
142334556778899910101011111112121212
1523345667788991010111111121212121313
16234456678899101011111112121213131314
172344567789910101111111212131313141414
182345567889910101111121213131314141415
1923456678891010111112121313131414151515
2023456678991010111212131313141415151516
21234567789101011111212131314141515161616
22234567889101011121213131414151516161617
23234567889101111121213141415151616161717
24234567899101111121313141415151616171718
252345678910101112121314141515161617171818
bound upper — reject if runs ≥
Runs Test Critical Values — upper — reject if runs ≥
n₁ / n₂2345678910111213141516171819202122232425
2————————————————————————
3————————————————————————
4———99———————————————————
5——910101111—————————————————
6——91011121213131313—————————————
7———1112131314141414151515——————————
8———11121314141515161616161717171717—————
9————1314141516161617171818181818181919191919
10————1314151616171718181819191920202020202020
11————1314151617171819191920202021212122222222
12————1314161617181919202021212122222222232323
13—————15161718191920202121222223232324242424
14—————15161718192020212222232323242424252525
15—————15161818192021222223232424252525262626
16——————171819202121222323242525252626272727
17——————171819202122232324252526262727272828
18——————171819202122232425252626272728282929
19——————171820212223232425262627272829292930
20——————171820212223242525262727282929303031
21———————1920212223242526272728292930303131
22———————1920222224242526272829293030313132
23———————1920222324252627272829303031323233
24———————1920222324252627282929303131323333
25———————1920222324252627282930313132333334

A dash means no rejection is possible on that side at these sample sizes. The table is symmetric in n₁ and n₂, and runs to n = 25.

How these values were produced

Exact distribution of the number of runs in a binary sequence. Tabulated by Swed & Eisenhart (1943), Ann. Math. Statist. 14(1), 66–87.

Every one of these 1,152 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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