F-Distribution Table

The F-table gives critical values of the F-distribution, used in ANOVA and to compare variances. Rows are the denominator degrees of freedom (df₂), columns the numerator degrees of freedom (df₁). Two tables are shown for the two most common significance levels.

Example: a one-way ANOVA with df₁ = 3, df₂ = 20 at α = 0.05 → critical F = 3.10. Reject H₀ if your F-statistic exceeds this.

How to read the F-table

Find the numerator degrees of freedom df₁ (the column). In a one-way ANOVA with k groups, df₁ = k − 1.
Find the denominator degrees of freedom df₂ (the row). In a one-way ANOVA with N observations in total, df₂ = N − k.
Use the table for your significance level (α = 0.05 or 0.01 above). If your F statistic is larger than the cell value, the group means are not all equal.

Worked example: one-way ANOVA

Four teaching methods are compared with 10 students each, so N = 40 and k = 4. The degrees of freedom are df₁ = 3 and df₂ = 36, and the ANOVA gives F = 3.10. Row 36 is not in the table, so use the next smaller row, 30: the critical value at α = 0.05 is 2.922. Since 3.10 > 2.922, at least one method's mean differs from the others. At α = 0.01 the critical value is 4.510, so the result is not significant at the 1% level. (The exact critical value for df₂ = 36 is 2.866 and the exact p-value is 0.039; the ANOVA calculator gives both.)

A significant F only says that the means are not all equal. To find which groups differ, follow up with a post-hoc test such as Tukey's HSD (see the Tukey HSD table).

Other uses of the F-table

Regression: the overall F-test uses df₁ = number of predictors and df₂ = n − predictors − 1.
Comparing two variances: put the larger sample variance on top; df₁ and df₂ are the two sample sizes minus one. For a two-sided test at 5%, use the 0.025 critical value rather than the 0.05 table.

Common mistakes

Swapping df₁ and df₂. F(3, 30) and F(30, 3) are very different distributions. Numerator df go across the top, denominator df down the side.
Rounding df₂ up to the next row. Rounding up gives a smaller critical value and can turn a non-significant result into a significant one. Round down.
Stopping at a significant F. Report which groups differ (post-hoc tests) and how much (effect size such as η²).

Need more precision than a table gives? The p-value calculator returns the exact p-value for any F and df, and how to calculate a p-value shows the method.

Frequently asked questions

How do you read an F-table?

Use the column for the numerator degrees of freedom (df1) and the row for the denominator degrees of freedom (df2), in the table for your significance level. If your F statistic is larger than that value, the result is significant.

What are df1 and df2 in a one-way ANOVA?

df1 is the number of groups minus 1, and df2 is the total number of observations minus the number of groups. Four groups of 10 give df1 = 3 and df2 = 36.

What if df2 is not in the F-table?

Use the next smaller df2 in the table. Its critical value is slightly larger, so the conclusion stays conservative.

Does a significant F-test tell me which groups differ?

No. It only shows that not all group means are equal. Use a post-hoc test such as Tukey's HSD to find which pairs differ.

α = 0.05

df₂ \ df₁	1	2	3	4	5	6	7	8	9	10	12	15	20	24	30	60
1	161.45	199.50	215.71	224.58	230.16	233.99	236.77	238.88	240.54	241.88	243.91	245.95	248.01	249.05	250.10	252.20
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	19.48
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	8.57
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	5.69
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	4.43
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	3.74
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	3.30
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	3.01
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	2.79
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	2.62
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	2.49
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	2.38
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	2.30
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	2.22
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	2.16
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	2.11
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	2.06
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	2.02
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	1.98
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	1.95
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	1.84
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	1.74
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	1.64
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	1.53
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	1.43

α = 0.01

df₂ \ df₁	1	2	3	4	5	6	7	8	9	10	12	15	20	24	30	60
1	4052.18	4999.50	5403.35	5624.58	5763.65	5858.99	5928.36	5981.07	6022.47	6055.85	6106.32	6157.28	6208.73	6234.63	6260.65	6313.03
2	98.50	99.00	99.17	99.25	99.30	99.33	99.36	99.37	99.39	99.40	99.42	99.43	99.45	99.46	99.47	99.48
3	34.12	30.82	29.46	28.71	28.24	27.91	27.67	27.49	27.35	27.23	27.05	26.87	26.69	26.60	26.50	26.32
4	21.20	18.00	16.69	15.98	15.52	15.21	14.98	14.80	14.66	14.55	14.37	14.20	14.02	13.93	13.84	13.65
5	16.26	13.27	12.06	11.39	10.97	10.67	10.46	10.29	10.16	10.05	9.89	9.72	9.55	9.47	9.38	9.20
6	13.75	10.92	9.78	9.15	8.75	8.47	8.26	8.10	7.98	7.87	7.72	7.56	7.40	7.31	7.23	7.06
7	12.25	9.55	8.45	7.85	7.46	7.19	6.99	6.84	6.72	6.62	6.47	6.31	6.16	6.07	5.99	5.82
8	11.26	8.65	7.59	7.01	6.63	6.37	6.18	6.03	5.91	5.81	5.67	5.52	5.36	5.28	5.20	5.03
9	10.56	8.02	6.99	6.42	6.06	5.80	5.61	5.47	5.35	5.26	5.11	4.96	4.81	4.73	4.65	4.48
10	10.04	7.56	6.55	5.99	5.64	5.39	5.20	5.06	4.94	4.85	4.71	4.56	4.41	4.33	4.25	4.08
11	9.65	7.21	6.22	5.67	5.32	5.07	4.89	4.74	4.63	4.54	4.40	4.25	4.10	4.02	3.94	3.78
12	9.33	6.93	5.95	5.41	5.06	4.82	4.64	4.50	4.39	4.30	4.16	4.01	3.86	3.78	3.70	3.54
13	9.07	6.70	5.74	5.21	4.86	4.62	4.44	4.30	4.19	4.10	3.96	3.82	3.66	3.59	3.51	3.34
14	8.86	6.51	5.56	5.04	4.69	4.46	4.28	4.14	4.03	3.94	3.80	3.66	3.51	3.43	3.35	3.18
15	8.68	6.36	5.42	4.89	4.56	4.32	4.14	4.00	3.89	3.80	3.67	3.52	3.37	3.29	3.21	3.05
16	8.53	6.23	5.29	4.77	4.44	4.20	4.03	3.89	3.78	3.69	3.55	3.41	3.26	3.18	3.10	2.93
17	8.40	6.11	5.18	4.67	4.34	4.10	3.93	3.79	3.68	3.59	3.46	3.31	3.16	3.08	3.00	2.83
18	8.29	6.01	5.09	4.58	4.25	4.01	3.84	3.71	3.60	3.51	3.37	3.23	3.08	3.00	2.92	2.75
19	8.18	5.93	5.01	4.50	4.17	3.94	3.77	3.63	3.52	3.43	3.30	3.15	3.00	2.92	2.84	2.67
20	8.10	5.85	4.94	4.43	4.10	3.87	3.70	3.56	3.46	3.37	3.23	3.09	2.94	2.86	2.78	2.61
24	7.82	5.61	4.72	4.22	3.90	3.67	3.50	3.36	3.26	3.17	3.03	2.89	2.74	2.66	2.58	2.40
30	7.56	5.39	4.51	4.02	3.70	3.47	3.30	3.17	3.07	2.98	2.84	2.70	2.55	2.47	2.39	2.21
40	7.31	5.18	4.31	3.83	3.51	3.29	3.12	2.99	2.89	2.80	2.66	2.52	2.37	2.29	2.20	2.02
60	7.08	4.98	4.13	3.65	3.34	3.12	2.95	2.82	2.72	2.63	2.50	2.35	2.20	2.12	2.03	1.84
120	6.85	4.79	3.95	3.48	3.17	2.96	2.79	2.66	2.56	2.47	2.34	2.19	2.03	1.95	1.86	1.66

Computed with SciPy (f.ppf). Run a full analysis with the One-Way ANOVA Calculator.

All tables Z-Score Calculator T-Test Chi-Square Test Interpreting p-values

How to read the F-table

Worked example: one-way ANOVA

Other uses of the F-table

Common mistakes

Frequently asked questions

How do you read an F-table?

What are df1 and df2 in a one-way ANOVA?

What if df2 is not in the F-table?

Does a significant F-test tell me which groups differ?

α = 0.05

α = 0.01

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