F-Distribution Probability Calculator
The F-distribution arises whenever you compare two variances: in the analysis of variance, in variance-ratio tests, and in the overall significance test of a regression. This calculator takes an F statistic and the numerator and denominator degrees of freedom and returns the cumulative probability to the left, the right-tail p-value used to judge significance, and the percentile. The computation uses the regularized incomplete beta function, evaluated with a numerically stable continued fraction, so it is accurate across the wide range of degrees of freedom seen in real tests.
F-distribution formula
Let x = df1 * F / (df1 * F + df2)
CDF F(F; df1, df2) = I(df1/2, df2/2, x), the regularized incomplete beta
Right-tail p-value = 1 - CDF
Percentile = CDF * 100
The F statistic must be non-negative and both degrees of freedom must be positive.
F-test context
- ANOVA numerator df is the number of groups minus one; denominator df is total observations minus the number of groups.
- The numerator and denominator degrees of freedom are not interchangeable; order matters.
- The F-test rejection region is the right tail because the statistic is a non-negative variance ratio.
- An F of 6.591 at df1 = 3 and df2 = 4 gives a right-tail p of about 0.05.
- The square of a t statistic with df degrees of freedom follows an F-distribution with 1 and df degrees of freedom.
F-distribution probability: frequently asked questions
What does the F-distribution calculator return?
Enter an F statistic and its two degrees of freedom: the numerator (df1) and the denominator (df2). It returns the cumulative probability to the left of the statistic, the right-tail p-value used in ANOVA and variance-ratio tests, and the percentile.
How is the F-distribution probability computed?
The cumulative distribution function is the regularized incomplete beta function I(df1/2, df2/2, x), where x = df1 times F divided by (df1 times F plus df2). The right-tail p-value is one minus that value. The incomplete beta is evaluated with a stable continued-fraction expansion.
Where is the F-distribution used?
It is the reference distribution for the analysis of variance (ANOVA), for comparing two sample variances, and for testing the overall significance of a regression model. In each case the test statistic is a ratio of variances that follows an F-distribution under the null hypothesis.
Which degree of freedom is the numerator?
The numerator degrees of freedom (df1) come from the variance in the numerator of the F ratio, such as the between-groups variance in ANOVA. The denominator degrees of freedom (df2) come from the error or within-groups variance. Order matters, so enter them correctly.
Why is the F-test usually one-tailed on the right?
An F statistic is a ratio of variances and is always non-negative. Evidence against the null hypothesis appears as a large ratio, so the relevant rejection region is the right tail. That is why the right-tail p-value is the value most often reported.
Official sources
- U.S. NIST/SEMATECH e-Handbook of Statistical Methods: Engineering Statistics Handbook, F-distribution.
- NIST Digital Library of Mathematical Functions: Incomplete Beta and Gamma Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.