F-Distribution Table

How to Use the F-Distribution Table

1. 1. Choose α. Pick the significance level at the top. Each α has its own table, because F depends on two degrees of freedom at once.
2. 2. Find the two degrees of freedom. The numerator df (df₁) runs across the top and the denominator df (df₂) down the side. In a one-way ANOVA, df₁ = k − 1 for k groups and df₂ = N − k.
3. 3. Read the critical value. If your F statistic is at least that large, the result is significant at the chosen level.
4. 4. Use the lookup for anything else. Any df₁, df₂ and α — including df the table skips — plus the lower critical value for two-sided variance tests.

How to Read the F Table

F is the ratio of two variance estimates, so it carries two degrees of freedom: df₁ for the numerator and df₂ for the denominator. Each entry is the value with an area of α to its right. In a one-way ANOVA comparing 3 groups of 10, df₁ = 3 − 1 = 2 and df₂ = 30 − 3 = 27; at α = .05 the table gives 3.354, so an F of 3.35 just misses significance (p = .0502).

The F distribution is not symmetric, so the lower critical value is not simply the negative of the upper one. It comes from swapping the degrees of freedom: the lower-tail value F(α; df₁, df₂) equals 1 / F(α; df₂, df₁) from the upper tail — for the variance test above, 1 / 4.026 = 0.2484. The lookup box gives both.

What the ∞ Row and Column Mean

As a degrees-of-freedom value grows without bound, its variance estimate becomes exact, and F turns into a scaled chi-square: with df₂ = ∞, F = χ²(df₁) / df₁; with df₁ = ∞, F = df₂ / χ²(df₂). The table computes these limits exactly rather than plugging in a large number, and where both are infinite F = 1.

Common mistakes

• Swapping the degrees of freedom. F(2, 24) and F(24, 2) are very different — 3.403 against 19.454 at α = .05. The numerator df always goes across the top.
• Reading the wrong table. Each significance level is a separate table, so check the heading above the table before you read from it.
• Halving α for ANOVA. The ANOVA F test is right-tailed by construction: for α = .05 use the .05 table, not .025.

Frequently Asked Questions

How do you read an F table?

Choose the table for your α, go across the top to your numerator degrees of freedom (df₁) and down the side to your denominator degrees of freedom (df₂). The entry is the critical value: reject the null hypothesis if your F is at least that large. For df₁ = 5, df₂ = 10 and α = .05 it is 3.326.

What are the degrees of freedom for an F test in ANOVA?

For a one-way ANOVA with k groups and N observations in total, df₁ = k − 1 (between groups) and df₂ = N − k (within groups). For the overall F test of a regression with p predictors and n cases, df₁ = p and df₂ = n − p − 1.

Is the F test one-tailed or two-tailed?

The F tests in ANOVA and regression are right-tailed, because only a large ratio counts as evidence against the null hypothesis. A test of whether two variances are equal is usually two-sided and needs both the upper and the lower critical values.

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

Use the lookup box for the exact value — F(.05; 2, 35) = 3.2674, for instance. With only a printed table, the conservative choice is the next smaller df that appears, which gives a slightly larger critical value.

How are the F and t distributions related?

An F with one numerator degree of freedom is the square of a t: F(1, df) = t(df)². That is why F(.05; 1, 10) = 4.965 = 2.228², and why an ANOVA on two groups gives exactly the same p-value as a two-tailed independent t-test.

Why do F critical values approach 1 for large degrees of freedom?

When the null hypothesis is true, the numerator and denominator both estimate the same variance. With many degrees of freedom both estimates become precise, so their ratio settles near 1; the critical values shrink towards 1 as both df grow and equal exactly 1 when both are infinite.