How to Read the F-Distribution Table
The F-table uses two degrees of freedom instead of one — that is what makes it wider than a t-table or chi-square table. The numerator df (df₁) runs across the top: it comes from the treatment effect you’re testing, the number of predictors in a regression, or the between-group variance in ANOVA. The denominator df (df₂) runs down the left side: it comes from the residual or within-group variance.
To use the table: pick your α tab at the top, find the df₁ column across the header, trace down to the df₂ row, and read the critical value. If your F-statistic exceeds that number, the result is significant at that α level. The typed lookup above the table lets you enter any df₁ and df₂ — including values not listed in the printed grid — and shows critical values for all five α levels at once.
Where the Two df Values Come From
The split into two degrees of freedom is not a convention — the F-statistic is literally a ratio of two chi-square estimates, and each estimate has its own sample size. Understanding where each df comes from prevents the most common table-reading error: swapping df₁ and df₂.
In a one-way ANOVA with k groups and N total observations: df₁ = k − 1 (between-groups) and df₂ = N − k (within-groups, also called residual). For a two-sample F-test for equal variances: df₁ = n₁ − 1 and df₂ = n₂ − 1. For a regression F-test with p predictors: df₁ = p and df₂ = n − p − 1.
A Worked Example
Three study-skill conditions, ten students in each, so N = 30 and k = 3. An ANOVA yields F = 4.26. The degrees of freedom are df₁ = k − 1 = 2 and df₂ = N − k = 27.
Look up df₁ = 2 and df₂ = 27 in the α = 0.05 tab. The table interpolates to F* ≈ 3.354. Because 4.26 > 3.354, the null hypothesis — that the three conditions have equal means — is rejected at the 5% level. The exact p-value (from the lookup field) is roughly 0.024.
The interpretation: the between-condition variance is about 4.26 times larger than the within-condition noise. That ratio is too large to attribute to chance at α = 0.05. Had the study included only five students per group (N = 15, df₂ = 12), the critical value would have risen to 3.885 and F = 4.26 would still pass — but barely, and the p-value gap would have narrowed considerably.
The Connection to t and Chi-Square
When df₁ = 1, the F-statistic is the square of the corresponding t-statistic: F(1, ν) = t²(ν). The two-tailed t critical value at α = 0.05 for df = 20 is 2.086; the F-table entry at df₁ = 1, df₂ = 20 is 4.351, which is 2.086². This means a one-way ANOVA with exactly two groups gives the same decision as a pooled two-sample t-test.
The other direction: as df₂ grows, the F-distribution approaches chi-square(df₁)/df₁. At df₂ = 120, the F critical value at α = 0.05 with df₁ = 5 is already within about 3.5% of the chi-square critical value divided by 5. The three reference tables on this site — z, t, chi-square, and F — are not independent tools; they describe the same family of distributions from different vantage points.
Frequently Asked Questions
What is the F-distribution used for?
It appears wherever you compare two variance estimates: ANOVA, regression F-tests, Levene’s test for equal variances, and likelihood ratio tests in more complex models. The setup is the same in each case — a ratio of two chi-square estimates divided by their df. The F-table gives you the threshold that ratio must cross to call the result significant.
Why does the F-table need two degrees of freedom?
Because an F-statistic is a ratio of two independent estimates, and each estimate has its own df. The numerator df captures how many independent effects the treatment or model contributes; the denominator df captures how many observations are left over for the error estimate. Change either one and you change the shape of the null distribution.
What are df₁ and df₂ in a one-way ANOVA?
df₁ = k − 1 where k is the number of groups. df₂ = N − k where N is the total number of observations. For three groups of ten: df₁ = 2, df₂ = 27. For regression with p predictors: df₁ = p and df₂ = n − p − 1.
How do I get the p-value from an F-statistic?
Type df₁, df₂, and your F-statistic into the three fields above the table. The panel returns the upper-tail p immediately. For the ANOVA example above: F(2, 27) = 4.26 gives p ≈ 0.024.
What does F = 1 mean?
F = 1 means between-group variance equals within-group variance — signal matches noise. In ANOVA this is exactly what you expect when the treatment does nothing: all variation is sampling noise and the ratio of two noise estimates is, on average, 1. F = 1 never rejects the null at any standard α level.
How does the F-distribution relate to t and chi-square?
F(1, ν) = t²(ν) — two-group ANOVA and two-sample t-test are the same test. As df₂ → ∞, F(df₁, ∞) converges to χ²(df₁)/df₁. The z-distribution is t at df = ∞, which is the same as F(1, ∞). All three tables on this site are cross-sections of the same underlying distribution family.
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