Statistics How-To
How to Read an F Table
An F table gives the critical value of the F distribution for one significance level. Look up the numerator degrees of freedom across the top and the denominator degrees of freedom down the side, and read the number where they meet. If your F statistic is larger, the result is significant. For an ANOVA with 4 groups of 6 observations the degrees of freedom are 3 and 20 and the 5% critical value is 3.10.
Which degrees of freedom is which
| Test | Numerator df (across) | Denominator df (down) |
|---|---|---|
| One-way ANOVA, k groups, N observations | k − 1 | N − k |
| Regression with p predictors, n observations | p | n − p − 1 |
| Comparing two variances (larger over smaller) | n₁ − 1 for the larger variance | n₂ − 1 for the smaller variance |
Both numbers are degrees of freedom of a sum of squares divided by its count, so this is the place to review degrees of freedom if either is unclear.
The 5% table (α = 0.05)
| Denominator df | Num df 1 | Num df 2 | Num df 3 | Num df 4 | Num df 5 | Num df 10 |
|---|---|---|---|---|---|---|
| 3 | 10.13 | 9.55 | 9.28 | 9.12 | 9.01 | 8.79 |
| 5 | 6.61 | 5.79 | 5.41 | 5.19 | 5.05 | 4.74 |
| 10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 | 2.98 |
| 20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 | 2.35 |
| 30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 | 2.16 |
| 60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 | 1.99 |
For other significance levels, look at the same cell in the α = 0.025 and α = 0.01 tables: with 3 and 20 degrees of freedom the critical values are 3.10 (5%), 3.86 (2.5%) and 4.94 (1%). The F-table page has all four tables.
Reading it in four steps
- Compute F, the ratio of two mean squares (or of two variances).
- Work out the numerator and denominator degrees of freedom from the table above.
- Open the table for your α and find the column for the numerator df and the row for the denominator df.
- If your F is larger than the entry, reject the null hypothesis.
Worked examples
| Situation | F and df | Table value | Conclusion |
|---|---|---|---|
| ANOVA: 4 groups of 6 observations | F = 3.5, df 3 and 20 | 3.10 at α = 0.05 | Significant; exact p = 0.0345 |
| The same design, a smaller F | F = 2.9, df 3 and 20 | 3.10 at α = 0.05 | Not significant; exact p = 0.0603 |
| Regression with 1 predictor and 5 points | F = 4.5, df 1 and 3 | 10.13 at α = 0.05 | Not significant; exact p = 0.1240 |
In the last row an F of 4.5 looks large, but with only 3 denominator degrees of freedom the bar is high; it is the regression example from correlation and regression in Excel. The ANOVA calculator reports the exact p-value for any F.
Software and common mistakes
| Tool | Critical F, α = 0.05, df 3 and 20 | p-value of F = 3.5 |
|---|---|---|
| Excel and Google Sheets | =F.INV.RT(0.05,3,20) gives 3.0984 | =F.DIST.RT(3.5,3,20) gives 0.0345 |
| R | qf(0.95, 3, 20) gives 3.0984 | pf(3.5, 3, 20, lower.tail = FALSE) gives 0.0345 |
| Python (SciPy) | f.ppf(0.95, 3, 20) gives 3.0984 | f.sf(3.5, 3, 20) gives 0.0345 |
- Swapping the two df. The critical value for 3 and 20 degrees of freedom is 3.10, and for 20 and 3 it is 8.66. Always read numerator across, denominator down.
- Using a two-tailed rule for ANOVA. The F test for ANOVA and regression is right-tailed; use the α column as printed. Only the two-sided variance test uses α/2.
- Stopping at a significant F. It says some group differs, not which; follow up with Tukey HSD. The ANOVA explained guide covers the whole procedure.
Try the F-Table
Look up F critical values by numerator and denominator degrees of freedom for α = 0.10, 0.05, 0.025 and 0.01.
Try the ANOVA Calculator
Get the ANOVA table with F, both degrees of freedom and the p-value from your group data.
Frequently Asked Questions
Which degrees of freedom go across and which go down in an F table?
The numerator degrees of freedom go across the top and the denominator degrees of freedom go down the side. In a one-way ANOVA the numerator is the between-groups df, k − 1, and the denominator is the within-groups df, N − k. The order matters because the F distribution is not symmetric in the two.
Why are there separate F tables for each α?
The F distribution has two degrees-of-freedom parameters, so one table cannot also carry the tail area as a third dimension. Books print one full table for each α, usually 0.10, 0.05, 0.025 and 0.01. Software or the F-table page on this site lifts that limit for any α.
How is F related to t?
With one numerator degree of freedom, F is the square of t. The critical F for 1 and 10 degrees of freedom at α = 0.05 is 4.965, and the two-tailed critical t for 10 degrees of freedom is 2.228, whose square is 4.965. This is why a two-group ANOVA gives the same p-value as a pooled t-test.
How do I use the F table for a test of two variances?
Divide the larger sample variance by the smaller and compare the ratio with the critical F for (n₁ − 1, n₂ − 1) degrees of freedom. For a two-sided test at the 5% level use the α = 0.025 table. With variances 9 and 4 from samples of 10 and 15, F = 2.25, below the critical 3.209, so the difference is not significant.
What if my denominator degrees of freedom is not in the table?
Use the next smaller row, which is conservative, or compute it exactly. =F.INV.RT(0.05,2,27) in Excel gives 3.354, whereas the printed row for 30 gives 3.32 and the row for 20 gives 3.49. Rounding to the smaller row never turns a truly non-significant result into a significant one.
Does a significant F tell me which groups differ?
No. A significant F in an ANOVA says that at least one group mean differs from the others, not which. Follow it with a post hoc test such as Tukey's HSD, which compares the pairs while controlling the overall error rate.