F-Statistic Calculator
Calculate the F-statistic and p-value for variance comparison, ANOVA, or regression significance testing.
📖 What is the F-Statistic?
The F-statistic is a ratio of two variance estimates, named after the statistician Ronald A. Fisher. It is the foundation of three major statistical tests: the variance equality test (Snedecor's F), analysis of variance (ANOVA), and the regression model F-test. In all three contexts, the F-statistic answers a similar question: is the variance explained by a factor or model significantly larger than the unexplained (error) variance?
The F-distribution is right-skewed and bounded below by zero, defined by two parameters: the numerator degrees of freedom (df₁) and the denominator degrees of freedom (df₂). Large F values correspond to small p-values, indicating evidence against the null hypothesis. Unlike the t or Z distributions, the critical region for most F-tests is always in the upper tail - you reject H₀ when F is sufficiently large.
In one-way ANOVA, the F-statistic compares variance between group means (MS_between) to variance within groups (MS_within). If the groups truly have different means, between-group variance will be large relative to within-group variance, producing a large F. In regression, F compares the explained variance per predictor to the residual variance per degree of freedom - a significant F indicates the model outperforms a null model with no predictors.
This calculator handles all three F-test modes with full p-value calculation using the regularised incomplete beta function - the same mathematical approach used in professional statistical software.
📐 Formulas
Two-Variance F-Test: F = s₁² / s₂² - df₁ = n₁ − 1, df₂ = n₂ − 1 (always put larger s² in numerator; use two-sided p-value)
One-Way ANOVA: F = MS_between / MS_within - df₁ = k − 1, df₂ = n − k
- MS_between = SS_between / (k − 1) - mean square between groups
- MS_within = SS_within / (n − k) - mean square within groups (error)
- k = number of groups, n = total observations
Regression F-Test: F = (R² / k) / ((1 − R²) / (n − k − 1)) - df₁ = k, df₂ = n − k − 1
p-value: P(F(df₁, df₂) > F_observed) - computed via regularised incomplete beta function
Critical value F_crit: The value such that P(F > F_crit) = α. Reject H₀ if F > F_crit.
📖 How to Use This Calculator
📝 Example Calculations
Example 1 - Testing Equality of Process Variances
Example 2 - One-Way ANOVA: Three Drug Treatments
Example 3 - Regression Model Significance
Example 4 - ANOVA: Student Performance Across 4 Teaching Methods
❓ Frequently Asked Questions
🔗 Related Calculators
What is the F-statistic?
The F-statistic is a ratio of two variance estimates. It was developed by Ronald Fisher in the 1920s. In general, F = (variance explained by the model) / (variance not explained). A large F means the signal (explained variance) is large relative to the noise (unexplained variance), suggesting a real effect. The F-distribution is right-skewed, always positive, and depends on two degrees of freedom: df₁ (numerator) and df₂ (denominator).
When do I use the variance F-test?
Use the two-variance F-test (Snedecor's F) to test whether two populations have equal variances: H₀: σ₁² = σ₂². This is often done before a pooled t-test to check the equal-variance assumption, or in quality control to compare process consistency. F = s₁²/s₂². The test is sensitive to non-normality, so Levene's test is often preferred for this purpose in practice.
What is one-way ANOVA and how does the F-test work?
One-way ANOVA (Analysis of Variance) tests whether three or more group means are equal: H₀: μ₁ = μ₂ = ... = μ_k. It partitions total variance into SS_between (variance due to group differences) and SS_within (variance within groups/error). F = MS_between / MS_within = [SS_B/(k−1)] / [SS_W/(n−k)]. A significant F means at least one mean differs from the others.
What is the regression F-test?
The regression F-test tests whether the overall regression model explains a statistically significant proportion of variance in the dependent variable: H₀: all regression coefficients = 0 (model has no predictive power). F = (R²/k) / ((1−R²)/(n−k−1)), where R² is the coefficient of determination, k is the number of predictors, and n is the sample size. A significant F means the model is better than the null (intercept-only) model.
What are degrees of freedom for the F-test?
For two variances: df₁ = n₁ − 1, df₂ = n₂ − 1. For one-way ANOVA: df₁ = k − 1 (between groups), df₂ = n − k (within groups/error). For regression: df₁ = k (number of predictors), df₂ = n − k − 1 (residual). The degrees of freedom define the shape of the F-distribution and affect the critical value and p-value.
How do I find the critical F value?
The critical F value F_crit is the value such that P(F > F_crit) = α. If your computed F > F_crit, reject H₀. For ANOVA and regression F-tests, this is always a right-tail test (F is always positive; large F = evidence against H₀). For the two-variance test, it is two-sided. This calculator computes F_crit at your chosen α automatically.
What is the p-value for an F-test?
The p-value is P(F_distribution > F_observed). For ANOVA and regression, it is the probability of getting an F-statistic at least as large as observed if H₀ is true. A p-value < α (e.g., 0.05) means the result is statistically significant - reject H₀. For two-variance tests, the p-value is doubled (two-sided) since the test checks for equality in either direction.
What is the difference between ANOVA and multiple t-tests?
Running multiple t-tests between k groups leads to inflated Type I error rate. With k = 3 groups and α = 0.05, running 3 pairwise t-tests gives a family-wise error rate up to 1 − (1−0.05)³ ≈ 14%. ANOVA controls the family-wise Type I error at α for the omnibus test (are any means different?). After a significant ANOVA, post-hoc tests with corrections (Tukey, Bonferroni) are used for pairwise comparisons.
What does a non-significant F-test mean?
A non-significant F (p ≥ α) means you fail to reject H₀ - there is insufficient evidence that the groups differ (ANOVA), that the variances differ (variance test), or that the model is better than chance (regression). It does NOT prove H₀ is true. Low power (small n or small effect size) can lead to non-significant F even when a true difference exists.