Statistical Tables/Runs Test Critical Values

Hypothesis Testing

Runs Test Critical Values

The exact lower and upper number of runs that counts as evidence a sequence isn't random.

Hover or tap a cell to read off its lower–upper critical range.

Critical run counts (lower–upper), α=0.05 two-tailed — n₁ across, n₂ down

n₂ \ n₁	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	—	—	—	2–8	2–8	2–8	2–8	2–8	2–8	2–8	2–8	3–8	3–8	3–8	3–8	3–8	3–8	3–8
4	—	—	2–9	2–9	2–10	3–10	3–10	3–10	3–10	3–10	3–10	3–10	3–10	4–10	4–10	4–10	4–10	4–10
5	—	2–9	2–10	3–10	3–11	3–11	3–12	3–12	4–12	4–12	4–12	4–12	4–12	4–12	4–12	5–12	5–12	5–12
6	2–8	2–9	3–10	3–11	3–12	3–12	4–13	4–13	4–13	4–13	5–14	5–14	5–14	5–14	5–14	5–14	6–14	6–14
7	2–8	2–10	3–11	3–12	3–13	4–13	4–14	5–14	5–14	5–14	5–15	5–15	6–15	6–16	6–16	6–16	6–16	6–16
8	2–8	3–10	3–11	3–12	4–13	4–14	5–14	5–15	5–15	6–16	6–16	6–16	6–16	6–17	7–17	7–17	7–17	7–17
9	2–8	3–10	3–12	4–13	4–14	5–14	5–15	5–16	6–16	6–16	6–17	7–17	7–18	7–18	7–18	8–18	8–18	8–18
10	2–8	3–10	3–12	4–13	5–14	5–15	5–16	6–16	6–17	7–17	7–18	7–18	7–18	8–19	8–19	8–19	8–20	9–20
11	2–8	3–10	4–12	4–13	5–14	5–15	6–16	6–17	7–17	7–18	7–19	8–19	8–19	8–20	9–20	9–20	9–21	9–21
12	2–8	3–10	4–12	4–13	5–14	6–16	6–16	7–17	7–18	7–19	8–19	8–20	8–20	9–21	9–21	9–21	10–22	10–22
13	2–8	3–10	4–12	5–14	5–15	6–16	6–17	7–18	7–19	8–19	8–20	9–20	9–21	9–21	10–22	10–22	10–23	10–23
14	3–8	3–10	4–12	5–14	5–15	6–16	7–17	7–18	8–19	8–20	9–20	9–21	9–22	10–22	10–23	10–23	11–23	11–24
15	3–8	3–10	4–12	5–14	6–15	6–16	7–18	7–18	8–19	8–20	9–21	9–22	10–22	10–23	11–23	11–24	11–24	12–25
16	3–8	4–10	4–12	5–14	6–16	6–17	7–18	8–19	8–20	9–21	9–21	10–22	10–23	11–23	11–24	11–25	12–25	12–25
17	3–8	4–10	4–12	5–14	6–16	7–17	7–18	8–19	9–20	9–21	10–22	10–23	11–23	11–24	11–25	12–25	12–26	13–26
18	3–8	4–10	5–12	5–14	6–16	7–17	8–18	8–19	9–20	9–21	10–22	10–23	11–24	11–25	12–25	12–26	13–26	13–27
19	3–8	4–10	5–12	6–14	6–16	7–17	8–18	8–20	9–21	10–22	10–23	11–23	11–24	12–25	12–26	13–26	13–27	13–27
20	3–8	4–10	5–12	6–14	6–16	7–17	8–18	9–20	9–21	10–22	10–23	11–24	12–25	12–25	13–26	13–27	13–27	14–28

n₁ and n₂ are the counts of each of the two outcome types in your sequence; the table is symmetric in n₁ and n₂. A dash means no critical value is attainable at α=0.05 with so few observations. Reject the hypothesis of randomness if your observed number of runs R is at or below the lower value or at or above the upper value. Every value is the exact solution from the closed-form null distribution of the number of runs (Wald & Wolfowitz, 1940) — verified by brute-force enumeration of every possible ordering for several sample sizes.

How to use this tool

Count n₁, the number of one outcome type, and n₂, the number of the other, in your sequence.
Count R, the number of runs — unbroken streaks of the same outcome.
Find your n₁ and n₂ in the table to get the lower–upper critical range.
Reject randomness if R is at or below the lower value, or at or above the upper value.

Formula Reference

Exact PMF (even r = 2k)

P(R=2k) = 2·C(n₁−1,k−1)·C(n₂−1,k−1) / C(n₁+n₂,n₁)

Exact PMF (odd r = 2k+1)

P(R=2k+1) = [C(n₁−1,k−1)·C(n₂−1,k) + C(n₁−1,k)·C(n₂−1,k−1)] / C(n₁+n₂,n₁)

ArticleThe Runs Test: Checking Whether a Sequence Is Actually Random ArticleControl Charts GlossaryGlossary: Runs Test

Frequently Asked Questions

Why does the table only go up to n = 20?+

The exact distribution works for any sample size, but beyond n₁, n₂ ≈ 20 the runs count is closely normally distributed and a z-test based on the exact mean and variance of R is both fast and highly accurate — which is what most software defaults to at that point.

Why are some cells a dash?+

With very few observations of one type, even the most extreme possible arrangement isn't unusual enough to be significant at α=0.05 — the same situation the Spearman and runs tables both show for their smallest sample sizes.

What if my n1 and n2 aren't in the table?+

The table is symmetric — n₁=8, n₂=4 gives the same critical values as n₁=4, n₂=8, so check both orderings. For combinations larger than 20, use the normal approximation for R noted in the linked article.

Can I use this on continuous data?+

Yes — a common approach converts continuous data into two categories by comparing each value to the sample median, recording above or below, and running the test on that above/below sequence.

n₂ \ n₁

—

2–8

3–8

—

2–9

2–10

3–10

4–10

—

2–9

2–10

3–10

3–11

3–12

4–12

5–12

2–8

2–9

3–10

3–11

3–12

4–13

5–14

6–14

2–8

2–10

3–11

3–12

3–13

4–13

4–14

5–14

5–15

6–15

6–16

2–8

3–10

3–11

3–12

4–13

4–14

5–14

5–15

6–16

6–17

7–17

2–8

3–10

3–12

4–13

4–14

5–14

5–15

5–16

6–16

6–17

7–17

7–18

8–18

2–8

3–10

3–12

4–13

5–14

5–15

5–16

6–16

6–17

7–17

7–18

8–19

8–20

9–20

2–8

3–10

4–12

4–13

5–14

5–15

6–16

6–17

7–17

7–18

7–19

8–19

8–20

9–20

9–21

2–8

3–10

4–12

4–13

5–14

6–16

7–17

7–18

7–19

8–19

8–20

9–21

10–22

2–8

3–10

4–12

5–14

5–15

6–16

6–17

7–18

7–19

8–19

8–20

9–20

9–21

10–22

10–23

3–8

3–10

4–12

5–14

5–15

6–16

7–17

7–18

8–19

8–20

9–20

9–21

9–22

10–22

10–23

11–23

11–24

3–8

3–10

4–12

5–14

6–15

6–16

7–18

8–19

8–20

9–21

9–22

10–22

10–23

11–23

11–24

12–25

3–8

4–10

4–12

5–14

6–16

6–17

7–18

8–19

8–20

9–21

10–22

10–23

11–23

11–24

11–25

12–25

3–8

4–10

4–12

5–14

6–16

7–17

7–18

8–19

9–20

9–21

10–22

10–23

11–23

11–24

11–25

12–25

12–26

13–26

3–8

4–10

5–12

5–14

6–16

7–17

8–18

8–19

9–20

9–21

10–22

10–23

11–24

11–25

12–25

12–26

13–26

13–27

3–8

4–10

5–12

6–14

6–16

7–17

8–18

8–20

9–21

10–22

10–23

11–23

11–24

12–25

12–26

13–26

13–27

3–8

4–10

5–12

6–14

6–16

7–17

8–18

9–20

9–21

10–22

10–23

11–24

12–25

13–26

13–27

14–28

How to use this tool

Count n₁, the number of one outcome type, and n₂, the number of the other, in your sequence.

Count R, the number of runs — unbroken streaks of the same outcome.

Find your n₁ and n₂ in the table to get the lower–upper critical range.

Reject randomness if R is at or below the lower value, or at or above the upper value.

Frequently Asked Questions

Why does the table only go up to n = 20?+

Why are some cells a dash?+

What if my n1 and n2 aren't in the table?+

Can I use this on continuous data?+

Yes — a common approach converts continuous data into two categories by comparing each value to the sample median, recording above or below, and running the test on that above/below sequence.