Statistics Reference

F-Table (F-Distribution Critical Values)

The F-table lists upper-tail critical values of the F distribution: the cutoffs an F statistic must exceed to be significant. Columns are numerator degrees of freedom (df₁), rows are denominator degrees of freedom (df₂), and each significance level α gets its own table. The workhorse α = 0.05 table is printed in full below, and each of the four standard levels has its own dedicated page.

Tables by Significance Level

α = 0.10 (90% confidence)

Exploratory or screening ANOVA where follow-up testing will confirm findings

α = 0.05 (95% confidence)

One-way and factorial ANOVA at the conventional 5% significance level

α = 0.025 (97.5% confidence)

Two-tailed tests of equal variances at the overall 5% level (0.025 per tail)

α = 0.01 (99% confidence)

ANOVA and regression F-tests reported as significant at the 1% level

How to Read the F-Table

You need three things: the numerator degrees of freedom, the denominator degrees of freedom, and the significance level. Pick the table for your α, then find the cell where the df₁ column meets the df₂ row.

Example: one-way ANOVA with 3 groups of 8 observations

Numerator df: df₁ = k − 1 = 3 − 1 = 2 (between groups).
Denominator df: df₂ = N − k = 24 − 3 = 21 (within groups).
In the α = 0.05 table, column 2 meets row 21 at 3.47. Reject the null hypothesis of equal means when F exceeds 3.47.

To run the whole procedure instead of looking up the cutoff, use the ANOVA calculator, which reports the F statistic together with its exact p-value.

F Critical Values, Upper-Tail α = 0.05

The table used by standard ANOVA and regression F-tests at the conventional 5% level. The ∞ row and column are computed from the exact chi-square limits of the F distribution. Values are rounded to two decimals (fewer for the very large small-df entries). Need another level? Open the dedicated 0.10, 0.025, or 0.01 tables.

Upper-tail F critical values at alpha = 0.05 by numerator and denominator degrees of freedom
df₂ \ df₁	1	2	3	4	5	6	7	8	9	10	12	15	20	24	30	40	60	120	∞
1	161.4	199.5	215.7	224.6	230.2	234.0	236.8	238.9	240.5	241.9	243.9	245.9	248.0	249.1	250.1	251.1	252.2	253.3	254.3
2	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38	19.40	19.41	19.43	19.45	19.45	19.46	19.47	19.48	19.49	19.50
3	10.13	9.55	9.28	9.12	9.01	8.94	8.89	8.85	8.81	8.79	8.74	8.70	8.66	8.64	8.62	8.59	8.57	8.55	8.53
4	7.71	6.94	6.59	6.39	6.26	6.16	6.09	6.04	6.00	5.96	5.91	5.86	5.80	5.77	5.75	5.72	5.69	5.66	5.63
5	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77	4.74	4.68	4.62	4.56	4.53	4.50	4.46	4.43	4.40	4.36
6	5.99	5.14	4.76	4.53	4.39	4.28	4.21	4.15	4.10	4.06	4.00	3.94	3.87	3.84	3.81	3.77	3.74	3.70	3.67
7	5.59	4.74	4.35	4.12	3.97	3.87	3.79	3.73	3.68	3.64	3.57	3.51	3.44	3.41	3.38	3.34	3.30	3.27	3.23
8	5.32	4.46	4.07	3.84	3.69	3.58	3.50	3.44	3.39	3.35	3.28	3.22	3.15	3.12	3.08	3.04	3.01	2.97	2.93
9	5.12	4.26	3.86	3.63	3.48	3.37	3.29	3.23	3.18	3.14	3.07	3.01	2.94	2.90	2.86	2.83	2.79	2.75	2.71
10	4.96	4.10	3.71	3.48	3.33	3.22	3.14	3.07	3.02	2.98	2.91	2.85	2.77	2.74	2.70	2.66	2.62	2.58	2.54
11	4.84	3.98	3.59	3.36	3.20	3.09	3.01	2.95	2.90	2.85	2.79	2.72	2.65	2.61	2.57	2.53	2.49	2.45	2.40
12	4.75	3.89	3.49	3.26	3.11	3.00	2.91	2.85	2.80	2.75	2.69	2.62	2.54	2.51	2.47	2.43	2.38	2.34	2.30
13	4.67	3.81	3.41	3.18	3.03	2.92	2.83	2.77	2.71	2.67	2.60	2.53	2.46	2.42	2.38	2.34	2.30	2.25	2.21
14	4.60	3.74	3.34	3.11	2.96	2.85	2.76	2.70	2.65	2.60	2.53	2.46	2.39	2.35	2.31	2.27	2.22	2.18	2.13
15	4.54	3.68	3.29	3.06	2.90	2.79	2.71	2.64	2.59	2.54	2.48	2.40	2.33	2.29	2.25	2.20	2.16	2.11	2.07
16	4.49	3.63	3.24	3.01	2.85	2.74	2.66	2.59	2.54	2.49	2.42	2.35	2.28	2.24	2.19	2.15	2.11	2.06	2.01
17	4.45	3.59	3.20	2.96	2.81	2.70	2.61	2.55	2.49	2.45	2.38	2.31	2.23	2.19	2.15	2.10	2.06	2.01	1.96
18	4.41	3.55	3.16	2.93	2.77	2.66	2.58	2.51	2.46	2.41	2.34	2.27	2.19	2.15	2.11	2.06	2.02	1.97	1.92
19	4.38	3.52	3.13	2.90	2.74	2.63	2.54	2.48	2.42	2.38	2.31	2.23	2.16	2.11	2.07	2.03	1.98	1.93	1.88
20	4.35	3.49	3.10	2.87	2.71	2.60	2.51	2.45	2.39	2.35	2.28	2.20	2.12	2.08	2.04	1.99	1.95	1.90	1.84
21	4.32	3.47	3.07	2.84	2.68	2.57	2.49	2.42	2.37	2.32	2.25	2.18	2.10	2.05	2.01	1.96	1.92	1.87	1.81
22	4.30	3.44	3.05	2.82	2.66	2.55	2.46	2.40	2.34	2.30	2.23	2.15	2.07	2.03	1.98	1.94	1.89	1.84	1.78
23	4.28	3.42	3.03	2.80	2.64	2.53	2.44	2.37	2.32	2.27	2.20	2.13	2.05	2.01	1.96	1.91	1.86	1.81	1.76
24	4.26	3.40	3.01	2.78	2.62	2.51	2.42	2.36	2.30	2.25	2.18	2.11	2.03	1.98	1.94	1.89	1.84	1.79	1.73
25	4.24	3.39	2.99	2.76	2.60	2.49	2.40	2.34	2.28	2.24	2.16	2.09	2.01	1.96	1.92	1.87	1.82	1.77	1.71
26	4.23	3.37	2.98	2.74	2.59	2.47	2.39	2.32	2.27	2.22	2.15	2.07	1.99	1.95	1.90	1.85	1.80	1.75	1.69
27	4.21	3.35	2.96	2.73	2.57	2.46	2.37	2.31	2.25	2.20	2.13	2.06	1.97	1.93	1.88	1.84	1.79	1.73	1.67
28	4.20	3.34	2.95	2.71	2.56	2.45	2.36	2.29	2.24	2.19	2.12	2.04	1.96	1.91	1.87	1.82	1.77	1.71	1.65
29	4.18	3.33	2.93	2.70	2.55	2.43	2.35	2.28	2.22	2.18	2.10	2.03	1.94	1.90	1.85	1.81	1.75	1.70	1.64
30	4.17	3.32	2.92	2.69	2.53	2.42	2.33	2.27	2.21	2.16	2.09	2.01	1.93	1.89	1.84	1.79	1.74	1.68	1.62
40	4.08	3.23	2.84	2.61	2.45	2.34	2.25	2.18	2.12	2.08	2.00	1.92	1.84	1.79	1.74	1.69	1.64	1.58	1.51
60	4.00	3.15	2.76	2.53	2.37	2.25	2.17	2.10	2.04	1.99	1.92	1.84	1.75	1.70	1.65	1.59	1.53	1.47	1.39
120	3.92	3.07	2.68	2.45	2.29	2.18	2.09	2.02	1.96	1.91	1.83	1.75	1.66	1.61	1.55	1.50	1.43	1.35	1.25
∞	3.84	3.00	2.60	2.37	2.21	2.10	2.01	1.94	1.88	1.83	1.75	1.67	1.57	1.52	1.46	1.39	1.32	1.22	1.00

Degrees of Freedom for Common F-Tests

One-way ANOVA: df₁ = k − 1 and df₂ = N − k for k groups and N total observations.
Regression overall F-test: df₁ = p (number of predictors) and df₂ = n − p − 1.
Two-variance ratio test: df₁ = n₁ − 1 and df₂ = n₂ − 1, with the larger sample variance conventionally placed in the numerator.

Lower Tail and the Reciprocal Rule

Printed F-tables only carry the upper tail because the lower tail follows from a reciprocal identity: the lower-tail critical value for (df₁, df₂) is one divided by the upper-tail value for the swapped pair (df₂, df₁).

F(1 − α; df₁, df₂) = 1 / F(α; df₂, df₁)

Example: the lower 0.05 cutoff for F(5, 10) equals 1 / F(0.05; 10, 5) = 1 / 4.74 ≈ 0.211. A two-tailed variance-ratio test at 5% therefore rejects when the ratio falls below the reciprocal cutoff or above the upper value from the α = 0.025 table.

Where the F Distribution Comes From

An F variable is the ratio of two independent chi-square variables, each divided by its degrees of freedom. That construction is exactly what a variance ratio is: each sample variance is a scaled chi-square under normality. It also explains the infinity rows in the tables — as one chi-square's degrees of freedom grow, that side of the ratio settles at 1, and the F distribution collapses to a scaled chi-square. The ∞ entries here are computed from those exact limits rather than from a large stand-in value.

Two familiar special cases: F(1, df) is the square of a t variable with df degrees of freedom (compare the α = 0.05 column 1 against squared entries of the two-tail 0.05 column of the t-table), and df₁·F(df₁, ∞) is a chi-square with df₁ degrees of freedom.

Frequently Asked Questions

Which degrees of freedom go on top and which on the side?

The numerator degrees of freedom (df1) run across the top and the denominator degrees of freedom (df2) run down the side. In a one-way ANOVA, df1 = k - 1 (groups minus one) belongs to the between-groups mean square in the numerator of F, and df2 = N - k (total observations minus groups) belongs to the within-groups mean square in the denominator. Swapping them gives a wrong critical value, so always match numerator to columns and denominator to rows.

Why does the F-table need a separate table for each alpha?

The F distribution has two shape parameters (df1 and df2), so a single printed table can only cover one tail area at a time - unlike the t-table, which spends its columns on alpha because it has just one degree-of-freedom parameter. This page shows the workhorse 0.05 table in full; the levels 0.10, 0.025, and 0.01 each have their own complete table linked below.

How do I find a lower-tail (left-tail) F critical value?

Use the reciprocal identity: the lower critical value with tail area alpha for (df1, df2) equals 1 divided by the upper critical value with tail area alpha for the swapped pair (df2, df1). For example, the lower 0.05 cutoff for F(5, 10) is 1 / F(0.05; 10, 5) = 1 / 4.74 = 0.211. This identity is why printed tables only carry the upper tail.

What does it mean that an F statistic exceeds the table value?

In ANOVA, F is the ratio of between-groups variance to within-groups variance. If the null hypothesis were true (all group means equal), F would hover around 1 and exceed the table value with probability alpha only by chance. An observed F beyond the critical value therefore says the between-groups signal is too large to attribute to chance at that significance level, and the null hypothesis is rejected.

What if my exact degrees of freedom are not in the table?

Round the denominator degrees of freedom down to the nearest listed row (and the numerator down to the nearest listed column) for a conservative critical value - the value will be slightly too large, never too small. For the exact number at any pair of degrees of freedom, use the critical value calculator on this site, which evaluates the same inverse F function the table was generated from.

Why do the values approach 1 toward the bottom-right of each table?

As both degrees of freedom grow, the two sample variances in the F ratio each estimate their population variance more precisely, so under the null hypothesis the ratio concentrates around 1. The infinity row and column are computed from the exact chi-square limits of the F distribution, and the bottom-right cell F(∞, ∞) equals exactly 1 at every significance level.