What is the F-statistic?

The F-statistic is the ratio of two mean square estimates: between-group variance divided by within-group variance. A larger F-statistic indicates greater evidence against the null hypothesis that all group means are equal.

What does the p-value mean here?

The p-value is the probability of obtaining an F-statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. It equals 1 minus the CDF: P(F > observed F | H0 true). A p-value below your significance level (e.g., 0.05) leads to rejecting H0.

What are numerator and denominator degrees of freedom?

In ANOVA with k groups and n total observations, the numerator degrees of freedom d1 = k − 1 and the denominator degrees of freedom d2 = n − k. In regression with p predictors and n observations, d1 = p and d2 = n − p − 1.

How is the F CDF computed here?

The F CDF is computed via the regularized incomplete beta function: CDF(F; d1, d2) = I(d1·F / (d1·F + d2), d1/2, d2/2). This calculator uses a continued-fraction (Lentz) algorithm for the regularized incomplete beta function, accurate to approximately 12 significant digits.

When does the F-distribution have a mean and variance?

The mean exists when d2 > 2 and equals d2 / (d2 − 2). The variance exists when d2 > 4 and equals 2·d2²·(d1+d2−2) / (d1·(d2−2)²·(d2−4)). For small d2 these statistics are undefined or infinite.

Can I use this for a one-tailed or two-tailed test?

The F-test is inherently one-tailed (right-tailed) because negative F-statistics are impossible — variance ratios are always non-negative. The p-value displayed is the right-tail probability P(F > observed F).

F-Distribution Calculator

Name: F-Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter an F-statistic and numerator/denominator degrees of freedom to compute the CDF and p-value instantly.

Input Parameters

F-Statistic (F > 0)

Numerator df (d1, groups − 1)

Denominator df (d2, n − groups)

Results

CDF — P(F ≤ observed) —

p-value — P(F > observed) —

PDF at F —

Mean (d2 > 2) —

Variance (d2 > 4) —

Mode (d1 > 2) —

Key Formulas

CDF: I(d1·F/(d1·F+d2), d1/2, d2/2)
p-value: 1 − CDF
PDF: F^(d1/2−1) · (d2+d1·F)^(−(d1+d2)/2) · K
Mean: d2 / (d2 − 2), when d2 > 2
Variance: 2d2²(d1+d2−2) / (d1(d2−2)²(d2−4))
Mode: (d1−2)/d1 · d2/(d2+2), when d1 > 2

PDF Curve

F(3, 20)

The vertical line marks the observed F-statistic. The shaded area is the p-value (right tail).

Common Presets

Significance Level Guide

α = 0.10

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Critical F

α = 0.05

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Critical F

α = 0.01

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Critical F

Summary

Enter an F-statistic and numerator/denominator degrees of freedom to compute the CDF and p-value instantly.

How it works

Enter the observed F-statistic (must be greater than 0).
Enter the numerator degrees of freedom (d1) — typically the number of groups minus 1.
Enter the denominator degrees of freedom (d2) — typically the total observations minus the number of groups.
Click Calculate or edit any field to see the CDF, p-value, PDF, mean, and variance.
Read the p-value to decide statistical significance at your chosen alpha level.
Use the preset buttons to explore common ANOVA and regression scenarios.

Use cases

Determine statistical significance in one-way and two-way ANOVA.
Evaluate the overall fit of a regression model (F-test for joint significance).
Compare variances between two populations with an F-test.
Compute p-values for ANOVA output when no software is available.
Teach and visualize the shape of the F-distribution for different degrees of freedom.
Verify statistical software output during quality control.
Assess whether group means differ significantly in experimental research.
Evaluate nested model comparisons in regression and mixed models.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu