Interactive F-Test Calculator
Enter degrees of freedom and an observed F-statistic to compute cumulative probability, p-values, and critical F values.
What is an F distribution calculator?
An F distribution calculator helps you evaluate probabilities and critical values for the F distribution, which appears in many statistical tests including ANOVA, regression model comparison, and tests of equal variances. Given two degrees of freedom and an F-statistic, you can quickly compute p-values and determine statistical significance.
Why the F distribution matters
The F distribution is the ratio of two scaled chi-square variables and is always nonnegative. It is right-skewed, especially with small degrees of freedom. As degrees of freedom increase, the distribution becomes less skewed.
- ANOVA: Tests whether at least one group mean differs from the others.
- Regression: Tests whether a model explains significantly more variance than a null model.
- Variance ratio tests: Compares two population variances.
How this calculator works
Inputs
- df1: numerator degrees of freedom
- df2: denominator degrees of freedom
- F statistic (x): observed test statistic
- α: significance level used for critical-value decisions
Outputs
- CDF P(F ≤ x): left-tail cumulative probability
- Right-tail p-value: often used directly in ANOVA and regression F-tests
- Critical F: threshold where right-tail area equals α
- Decision rule: reject or fail to reject H₀ for a right-tailed F-test
F distribution formula (conceptual)
If X ~ F(df1, df2), then the cumulative distribution is computed via the regularized incomplete beta function. In practice, this page uses a stable numerical implementation of gamma and beta-function routines to evaluate the CDF and invert it for critical values.
Example interpretation
Suppose df1 = 5, df2 = 20, observed F = 2.5, and α = 0.05. If the right-tail p-value is less than 0.05, you reject the null hypothesis for a right-tailed F-test. If it is greater than 0.05, you fail to reject. The reported upper critical value gives the same decision threshold in statistic space.
Common mistakes to avoid
- Swapping numerator and denominator degrees of freedom.
- Using a two-tailed rule when your analysis calls for a one-tailed F-test.
- Treating p-values as effect sizes (they are not).
- Ignoring model assumptions like independence and approximate normality of errors.
Quick FAQ
Is the F distribution ever negative?
No. F-values are always zero or positive.
Can I use this as an ANOVA p-value calculator?
Yes. If you already have the ANOVA F-statistic and corresponding degrees of freedom, the right-tail probability shown here is the p-value used in the overall F-test.
What does a large F-statistic mean?
Generally, larger F-values indicate stronger evidence against the null hypothesis, but interpretation depends on df1, df2, and your chosen α level.