F-Table (F-Distribution Critical Values)

Two degrees of freedom, in a specific order

An F distribution describes a ratio of independent scaled chi-square variables. Its shape depends on numerator df and denominator df. These are not interchangeable.

The printed grids below report upper-tail cutoffs c such that P(F_df₁,df₂ > c) = α. Read numerator df across the columns and denominator df down the rows. The interactive lookup above supports other values.

Upper-tail critical values at α = 0.05

Upper-tail critical values at α = 0.01

Values are rounded to four decimals. For df or α not displayed, use the interactive lookup or statistical software rather than substituting a nearby entry silently.

Worked ANOVA lookup

Three independent groups contain nine observations in total. Their between-group sum of squares is 24 and within-group sum of squares is 6.

1. Numerator df = k − 1 = 2.
2. Denominator df = N − k = 6.
3. Between-group mean square = 24/2 = 12.
4. Within-group mean square = 6/6 = 1.
5. F = 12/1 = 12.

At α = 0.05, F-critical for df (2, 6) is 5.1433. Since 12 > 5.1433, reject equal means under the classical ANOVA model; p = 0.0080.

The overall test establishes evidence that not all means are equal, not which pairs differ. Read the ANOVA lesson for the raw observations and follow-up analysis.

Match the statistic to the degrees of freedom

Here p is the number of regression predictors excluding the intercept. Repeated-measures ANOVA, Welch ANOVA, and other models can use different df; do not apply this table of rules to every F statistic.

ANOVA uses an upper-tail test

The classical equal-means null is rejected for a large between/within mean-square ratio. You do not halve α because group means can differ in either direction. F is nonnegative and its distribution is asymmetric; there is no ±F cutoff.

For two groups under the same pooled equal-variance model, the ANOVA F statistic equals the corresponding two-sided pooled t statistic squared. That identity does not make Welch’s t-test equivalent to every classical two-group ANOVA.

A two-sided variance-ratio test is different

If testing equality of two independent normal-population variances with F = s₁²/s₂² and fixed group order, split α between the lower and upper tails. An upper-tail-only 5% cutoff is not a two-sided 5% procedure.

The reciprocal identity helps find a lower cutoff:

$F_{\alpha/2}(df_1,df_2)=\frac1{F_{1-\alpha/2}(df_2,df_1)}$

The subscripts here are cumulative probabilities. Note that the df swap when taking the reciprocal. The classical variance-ratio F-test is sensitive to nonnormality; choose a variance procedure whose assumptions suit the data.

Check the model and report the difference

Before using a classical ANOVA F reference, assess independent units, group distributions, influential points, and common error variance. More observations do not fix dependence or unequal-variance problems automatically.

Report the group sizes and means, F, both df, p, and effect estimates or planned contrasts. A significant F does not prove causation or practical importance. The table is a reference for a model, not a substitute for selecting that model.

Practice

Four independent groups have 24 observations total. Which df pair belongs in the classical one-way ANOVA lookup?

References and next steps

NIST’s one-way ANOVA overview explains the variance decomposition. Continue with ANOVA, multiple regression, or the degrees of freedom reference. The site currently has no dedicated ANOVA calculator; the lesson shows the calculation in full.