Statistical Tables/Friedman Test Exact Critical Values

Hypothesis Testing

Friedman Test Exact Critical Values

The exact critical value of Fr for comparing small numbers of related conditions, computed exactly rather than by the chi-square approximation.

Exact critical Fr — k conditions, n subjects/blocks

k, n	α = 0.1	α = 0.05	α = 0.01
k=3, n=2	—	—	—
k=3, n=3	6.0000	6.0000	—
k=3, n=4	6.0000	6.5000	8.0000
k=3, n=5	5.2000	6.4000	8.4000
k=3, n=6	5.3333	7.0000	9.0000
k=3, n=7	5.4286	7.1429	8.8571
k=3, n=8	5.2500	6.2500	9.0000
k=3, n=9	5.5556	6.2222	9.5556
k=4, n=2	6.0000	6.0000	—
k=4, n=3	6.6000	7.4000	9.0000
k=4, n=4	6.3000	7.8000	9.6000

k is the number of related conditions (treatments); n is the number of subjects or blocks. A dash means no value of Fr is significant at that α with so few subjects. Reject the null hypothesis of no difference among conditions if your computed Fr is at or above the tabled value. For k or n outside this table, use the chi-square distribution with k−1 degrees of freedom as an approximation.

How to use this tool

Within each subject, rank the k conditions from 1 to k.
Sum the ranks for each condition across all n subjects, and compute Fr.
Look up your number of conditions (k) and subjects (n).
Reject the null hypothesis of no difference if Fr is at or above the tabled value.

Formula Reference

Friedman Fr statistic

Fr = [12 / (nk(k+1))] × Σ(Rⱼ²) − 3n(k+1)

Kendall's W (equivalent)

W = Fr / [n(k−1)], ranging from 0 (no agreement) to 1 (perfect)

Large-sample approximation

Fr is approximately χ² distributed with k−1 degrees of freedom

ArticleThe Friedman Test ArticleThe Kruskal-Wallis Test TableKruskal-Wallis Exact Critical Values TableChi-Square Table GlossaryGlossary: Friedman Test

Frequently Asked Questions

Why does the table stop at n=9 for k=3 and n=4 for k=4?+

The number of equally likely rank arrangements is (k!)ⁿ, which grows extremely fast — k=3, n=9 already means checking over 10 million arrangements, and k=4, n=5 would mean nearly 8 million just for that one row. Beyond this range, the chi-square approximation is fast and accurate.

Why are some cells a dash?+

With very few subjects, even perfect agreement across every subject isn't rare enough to be significant at that α — the same situation the Kruskal-Wallis and Spearman tables show for their smallest sample sizes.

How were these values computed?+

Every one of the (k!)ⁿ equally likely ways the n subjects could each independently rank the k conditions was generated directly, Fr was computed for each one, and the exact upper-tail critical value was read off the resulting distribution.

What if I have more than 4 conditions, or more subjects than the table covers?+

Use the chi-square approximation with k−1 degrees of freedom from the linked chi-square table — it becomes reliable well before exact enumeration becomes impractical.

k, n

α = 0.1

α = 0.05

α = 0.01

k=3, n=2

—

k=3, n=3

6.0000

—

k=3, n=4

6.0000

6.5000

8.0000

k=3, n=5

5.2000

6.4000

8.4000

k=3, n=6

5.3333

7.0000

9.0000

k=3, n=7

5.4286

7.1429

8.8571

k=3, n=8

5.2500

6.2500

9.0000

k=3, n=9

5.5556

6.2222

9.5556

k=4, n=2

6.0000

—

k=4, n=3

6.6000

7.4000

9.0000

k=4, n=4

6.3000

7.8000

9.6000

Frequently Asked Questions

Why does the table stop at n=9 for k=3 and n=4 for k=4?+

Why are some cells a dash?+

How were these values computed?+

What if I have more than 4 conditions, or more subjects than the table covers?+

Use the chi-square approximation with k−1 degrees of freedom from the linked chi-square table — it becomes reliable well before exact enumeration becomes impractical.