Reference · Statistics

F Distribution Table

Critical F values at four significance levels, with the numerator and denominator df you pick highlighted.

Significance level (upper tail)

α = 0.10

α = 0.05

α = 0.025

α = 0.01

Numerator df (df₁)

Between-groups df — the number of groups minus 1.

Denominator df (df₂)

Within-groups df — the total sample size minus the number of groups.

Critical F at α = 0.05, df₁ = 3, df₂ = 10

3.708

F distribution critical values at α = 0.05, numerator df across, denominator df down
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	18.513	19.000	19.164	19.247	19.296	19.330	19.353	19.371	19.385	19.396	19.413	19.429	19.446	19.454	19.462	19.471	19.479	19.487	19.496
3	10.128	9.552	9.277	9.117	9.013	8.941	8.887	8.845	8.812	8.786	8.745	8.703	8.660	8.639	8.617	8.594	8.572	8.549	8.526
4	7.709	6.944	6.591	6.388	6.256	6.163	6.094	6.041	5.999	5.964	5.912	5.858	5.803	5.774	5.746	5.717	5.688	5.658	5.628
5	6.608	5.786	5.409	5.192	5.050	4.950	4.876	4.818	4.772	4.735	4.678	4.619	4.558	4.527	4.496	4.464	4.431	4.398	4.365
6	5.987	5.143	4.757	4.534	4.387	4.284	4.207	4.147	4.099	4.060	4.000	3.938	3.874	3.841	3.808	3.774	3.740	3.705	3.669
7	5.591	4.737	4.347	4.120	3.972	3.866	3.787	3.726	3.677	3.637	3.575	3.511	3.445	3.410	3.376	3.340	3.304	3.267	3.230
8	5.318	4.459	4.066	3.838	3.687	3.581	3.500	3.438	3.388	3.347	3.284	3.218	3.150	3.115	3.079	3.043	3.005	2.967	2.928
9	5.117	4.256	3.863	3.633	3.482	3.374	3.293	3.230	3.179	3.137	3.073	3.006	2.936	2.900	2.864	2.826	2.787	2.748	2.707
10	4.965	4.103	3.708	3.478	3.326	3.217	3.135	3.072	3.020	2.978	2.913	2.845	2.774	2.737	2.700	2.661	2.621	2.580	2.538
11	4.844	3.982	3.587	3.357	3.204	3.095	3.012	2.948	2.896	2.854	2.788	2.719	2.646	2.609	2.570	2.531	2.490	2.448	2.404
12	4.747	3.885	3.490	3.259	3.106	2.996	2.913	2.849	2.796	2.753	2.687	2.617	2.544	2.505	2.466	2.426	2.384	2.341	2.296
13	4.667	3.806	3.411	3.179	3.025	2.915	2.832	2.767	2.714	2.671	2.604	2.533	2.459	2.420	2.380	2.339	2.297	2.252	2.206
14	4.600	3.739	3.344	3.112	2.958	2.848	2.764	2.699	2.646	2.602	2.534	2.463	2.388	2.349	2.308	2.266	2.223	2.178	2.131
15	4.543	3.682	3.287	3.056	2.901	2.790	2.707	2.641	2.588	2.544	2.475	2.403	2.328	2.288	2.247	2.204	2.160	2.114	2.066
16	4.494	3.634	3.239	3.007	2.852	2.741	2.657	2.591	2.538	2.494	2.425	2.352	2.276	2.235	2.194	2.151	2.106	2.059	2.010
17	4.451	3.592	3.197	2.965	2.810	2.699	2.614	2.548	2.494	2.450	2.381	2.308	2.230	2.190	2.148	2.104	2.058	2.011	1.960
18	4.414	3.555	3.160	2.928	2.773	2.661	2.577	2.510	2.456	2.412	2.342	2.269	2.191	2.150	2.107	2.063	2.017	1.968	1.917
19	4.381	3.522	3.127	2.895	2.740	2.628	2.544	2.477	2.423	2.378	2.308	2.234	2.155	2.114	2.071	2.026	1.980	1.930	1.878
20	4.351	3.493	3.098	2.866	2.711	2.599	2.514	2.447	2.393	2.348	2.278	2.203	2.124	2.082	2.039	1.994	1.946	1.896	1.843
21	4.325	3.467	3.072	2.840	2.685	2.573	2.488	2.420	2.366	2.321	2.250	2.176	2.096	2.054	2.010	1.965	1.916	1.866	1.812
22	4.301	3.443	3.049	2.817	2.661	2.549	2.464	2.397	2.342	2.297	2.226	2.151	2.071	2.028	1.984	1.938	1.889	1.838	1.783
23	4.279	3.422	3.028	2.796	2.640	2.528	2.442	2.375	2.320	2.275	2.204	2.128	2.048	2.005	1.961	1.914	1.865	1.813	1.757
24	4.260	3.403	3.009	2.776	2.621	2.508	2.423	2.355	2.300	2.255	2.183	2.108	2.027	1.984	1.939	1.892	1.842	1.790	1.733
25	4.242	3.385	2.991	2.759	2.603	2.490	2.405	2.337	2.282	2.236	2.165	2.089	2.007	1.964	1.919	1.872	1.822	1.768	1.711
26	4.225	3.369	2.975	2.743	2.587	2.474	2.388	2.321	2.265	2.220	2.148	2.072	1.990	1.946	1.901	1.853	1.803	1.749	1.691
27	4.210	3.354	2.960	2.728	2.572	2.459	2.373	2.305	2.250	2.204	2.132	2.056	1.974	1.930	1.884	1.836	1.785	1.731	1.672
28	4.196	3.340	2.947	2.714	2.558	2.445	2.359	2.291	2.236	2.190	2.118	2.041	1.959	1.915	1.869	1.820	1.769	1.714	1.654
29	4.183	3.328	2.934	2.701	2.545	2.432	2.346	2.278	2.223	2.177	2.104	2.027	1.945	1.901	1.854	1.806	1.754	1.698	1.638
30	4.171	3.316	2.922	2.690	2.534	2.421	2.334	2.266	2.211	2.165	2.092	2.015	1.932	1.887	1.841	1.792	1.740	1.683	1.622
40	4.085	3.232	2.839	2.606	2.449	2.336	2.249	2.180	2.124	2.077	2.003	1.924	1.839	1.793	1.744	1.693	1.637	1.577	1.509
60	4.001	3.150	2.758	2.525	2.368	2.254	2.167	2.097	2.040	1.993	1.917	1.836	1.748	1.700	1.649	1.594	1.534	1.467	1.389
120	3.920	3.072	2.680	2.447	2.290	2.175	2.087	2.016	1.959	1.910	1.834	1.750	1.659	1.608	1.554	1.495	1.429	1.352	1.254
∞	3.841	2.996	2.605	2.372	2.214	2.099	2.010	1.938	1.880	1.831	1.752	1.666	1.571	1.517	1.459	1.394	1.318	1.221	1.001

The F table is not symmetric: F(df₁, df₂) and F(df₂, df₁) are different numbers, so read the numerator df across the top and the denominator df down the side, never the other way round.

An F-table gives the cut-off an ANOVA F-statistic must beat to be significant. Numerator df runs across the top, denominator df down the side: with 3 groups and 30 observations you have df₁ = 2 and df₂ = 27, and the α = 0.05 critical value is 3.354.

What an F-statistic compares

F is a ratio of two variances. In a one-way ANOVA the numerator is the variance between group means and the denominator is the variance within groups. If the groups genuinely share a mean, both estimate the same underlying variance and the ratio hovers near 1. If the group means differ, the numerator inflates and F climbs, which is why the test only ever looks at the upper tail.

Because it is built from two variance estimates, the distribution needs two degrees-of-freedom values rather than one. For a one-way ANOVA the numerator df is the number of groups minus 1, and the denominator df is the total sample size minus the number of groups. Three groups with ten observations each gives df₁ = 3 − 1 = 2 and df₂ = 30 − 3 = 27.

The order is not interchangeable

F(2, 27) and F(27, 2) are completely different numbers — 3.354 against 19.459 at α = 0.05. The numerator df always goes across the top of the table and the denominator df down the side. Swapping them is the classic F-table mistake, and it usually produces a critical value so large that a real effect looks non-significant.

F = MSbetween ÷ MSwithin

each mean square is a sum of squares divided by its own degrees of freedom

1
Count both degrees of freedom. Comparing 3 teaching methods across 30 students gives df₁ = 3 − 1 = 2 and df₂ = 30 − 3 = 27.
2
Pick the significance level. ANOVA is nearly always run at α = 0.05; the table above also carries 0.10, 0.025 and 0.01.
3
Find the numerator df column. Read across the top row to df₁ = 2 — not down the side, which is the denominator.
4
Read down to the denominator df row. Follow the df₁ = 2 column down to df₂ = 27 to get the critical value 3.354.
5
Compare it with your F-statistic. An ANOVA producing F = 4.10 exceeds 3.354, so at least one group mean differs at the 5% level.

Critical F values for common ANOVA designs

Upper-tail cut-offs at α = 0.05 and α = 0.01, written as F(df₁, df₂).

df₁ (numerator)	df₂ (denominator)	α = 0.05	α = 0.01
1	20	4.351	8.096
2	10	4.103	7.559
2	12	3.885	6.927
2	27	3.354	5.488
3	12	3.490	5.953
3	16	3.239	5.292
3	20	3.098	4.938
4	25	2.759	4.177

What a significant F does and does not tell you

A significant F is an omnibus result: it says the group means are not all equal, without saying which pair differs. Finding that out needs a post-hoc comparison such as Tukey’s HSD, which controls the error rate across the many pairwise tests that follow. Reporting only that F was significant leaves the actual finding unstated.

The test also assumes independent observations, roughly normal residuals, and similar variances across groups. The variance assumption matters most when group sizes are unequal; Levene’s test checks it, and Welch’s ANOVA is the standard fallback when it fails. Finally, note that a two-tailed F-test for comparing two variances uses α/2 in the table, because the printed values are one-tailed cut-offs.

How do I find df₁ and df₂ for an ANOVA?

The numerator df₁ is the number of groups minus 1, and the denominator df₂ is the total number of observations minus the number of groups. Three groups totalling 30 observations give df₁ = 2 and df₂ = 27.

Which df goes across the top of an F-table?

The numerator degrees of freedom, df₁, runs across the top and the denominator df₂ runs down the side. The distribution is not symmetric in its two df values, so swapping them gives the wrong critical value.

Why is the F-test one-tailed?

F is a ratio of between-group to within-group variance, and only an unusually large ratio is evidence that the group means differ. A small F means the groups look alike, so the rejection region sits entirely in the upper tail.

What does an F value near 1 mean?

It means the variance between group means is about the same size as the variance within groups — exactly what you would expect if every group shared the same true mean. The further F rises above 1, the harder that explanation becomes.

The F was significant. Which groups differ?

The F-test does not say. It is an omnibus test, so follow it with a post-hoc procedure such as Tukey’s HSD or Bonferroni-corrected pairwise comparisons, which keep the overall error rate controlled across the extra tests.

Can I use an F-table to compare two variances?

Yes — put the larger sample variance in the numerator and use each sample size minus 1 as its df. Because the printed values are one-tailed, a two-tailed variance test at the 5% level reads the α = 0.025 column.

F Distribution Table

What an F-statistic compares

The order is not interchangeable

Critical F values for common ANOVA designs

What a significant F does and does not tell you

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