F Score to P Value Calculator

Convert F-statistics to p-values for ANOVA, regression, and variance ratio tests. Supports one‑tailed and two‑tailed tests with interactive F‑distribution visualization.

F‑distribution: Used to compare variances, in ANOVA, and in regression models. p‑value = probability of observing an F‑statistic as extreme as yours under the null hypothesis.

F=3.89, df1=3, df2=30
F=5.99, df1=2, df2=20
F=2.5, df1=5, df2=50
F=0.5, df1=4, df2=40
F=4.0, df1=1, df2=30
Computing...

Understanding the F‑distribution and p‑value

The F‑distribution is a continuous probability distribution that arises frequently in analysis of variance (ANOVA), regression analysis, and tests for equality of variances. It is defined by two parameters: numerator degrees of freedom (df1) and denominator degrees of freedom (df2).

Mathematical Definition:

For an F‑statistic F with df1 and df2, the p‑value depends on the alternative hypothesis:

  • Right‑tailed: p = P(Fdf1,df2 > F) — most common (e.g., ANOVA, overall F-test)
  • Left‑tailed: p = P(Fdf1,df2 < F) — used when testing if a variance is less than another
  • Two‑tailed: p = 2·min[P(F > F), P(F < F)] — for two-sided variance tests

Note: The F‑distribution is non‑negative and right‑skewed.

Common critical F‑values (α = 0.05, right‑tailed)

df1\df2 10 20 30 ∞
1 4.96 4.35 4.17 3.84
2 4.10 3.49 3.32 3.00
3 3.71 3.10 2.92 2.60
5 3.33 2.71 2.53 2.21

Interpretation Guidelines

1

Small p‑value (typically ≤ 0.05): Strong evidence against the null hypothesis. The result is statistically significant.

2

Large p‑value (> 0.05): Weak evidence against the null; fail to reject the null hypothesis.

3

Right‑tailed is default: In ANOVA and regression, the F‑test is almost always right‑tailed because we test if explained variance is significantly greater than unexplained variance.

Applications

  • ANOVA: Testing for differences among group means.
  • Regression overall F‑test: Testing if the model explains a significant amount of variance.
  • Equality of variances (F‑test): Comparing two population variances (often two‑tailed).
  • Lack‑of‑fit test: Assessing model adequacy.

Calculator features:

  • Uses jStat library for precise F‑distribution CDF and inverse CDF
  • Interactive graph with shaded tail area corresponding to your test type
  • Displays all three p‑values (right, left, two‑tailed) for complete insight
  • Adjustable p‑value format (decimal or scientific)
  • Shows critical F‑value for your selected test and α

Frequently Asked Questions

An F‑test is any statistical test where the test statistic follows an F‑distribution under the null hypothesis. Common examples include ANOVA, testing for equal variances, and testing overall significance in regression.

In most applications (ANOVA, regression), the null hypothesis assumes no effect, and the alternative is that the effect is present. This leads to larger F‑values under the alternative, so we reject only when F is sufficiently large (right tail).

Yes, F can be less than 1. It indicates that the explained variance is smaller than the unexplained variance, which often happens by chance. The p‑value for a right‑tailed test will be large (>0.5), indicating no evidence against the null.

It uses the jStat library, which implements high‑accuracy numerical approximations for the F‑distribution. Results match standard F‑tables and statistical software (R, SAS, SPSS).

As both df1 and df2 increase, the F‑distribution approaches a normal distribution. The calculator handles large degrees of freedom (e.g., 1000) accurately.