The F Distribution, Explained
After reading this you can find a p-value from an F statistic, look up a critical value for a chosen significance level, and read both results without confusing the two degrees of freedom.
What the F distribution is
The F distribution describes the ratio of two scaled sums of squares. If you divide one estimate of variance by another, and both estimates come from normal data, that ratio follows an F distribution. It shows up whenever you compare variability against variability rather than a value against a mean.
The distribution has two shape parameters, both positive: the numerator degrees of freedom d_1 and the denominator degrees of freedom d_2. It lives on [0, \infty), is right skewed, and never takes negative values because it is a ratio of squares. As d_2 grows large, the distribution concentrates near 1, which is the value you expect when the two variances are equal.
Here is the hook. In a one-way ANOVA with 4 groups and 40 observations, the F statistic has d_1 = 3 and d_2 = 36. If the group means are truly equal, F stays near 1. An observed value of 3.7 sits far out in the right tail, and the tail probability tells you how surprising that is.
When to use it, and when not
Reach for the F distribution when your test statistic is a ratio of variances or of mean squares. The three common cases:
- ANOVA: between-group mean square divided by within-group mean square.
- Comparing two variances: the ratio s_1^2 / s_2^2 from two independent samples.
- Overall regression significance: the F test that all slope coefficients are zero.
Do not use the F distribution for a single mean or a single proportion. For one mean with unknown variance, use the t distribution. For a goodness-of-fit or independence test on counts, use the chi-square distribution. The F distribution assumes the underlying data are normal; with heavy skew or strong outliers the tail probabilities can be badly off.
The F statistic equals the ratio of two independent chi-square variables, each divided by its own degrees of freedom. That is why F and chi-square appear together in the same tests, and why the square of a t variable with d_2 degrees of freedom is an F variable with d_1 = 1.
The formula and the intuition
If U and V are independent chi-square variables with d_1 and d_2 degrees of freedom, then
Each chi-square divided by its degrees of freedom is roughly an average of squared deviations, so U/d_1 and V/d_2 are two estimates of the same variance when the null hypothesis holds. Their ratio hovers around 1. When the effect you are testing is real, the numerator variance grows while the denominator stays put, and F climbs.
The probability density function is
Here x \ge 0 is the F value, \Gamma is the gamma function, and d_1, d_2 are the two degrees of freedom. You rarely evaluate this by hand. What matters is the tail area to the right of your observed F, which is the p-value, and the calculator computes it directly from the cumulative distribution.
The mean of the distribution is d_2 / (d_2 - 2) for d_2 \gt 2. With d_2 = 10 the mean is 10/8 = 1.25, slightly above 1 because of the right skew.
A worked example with the demo data
p-value for F = 3.71 with d₁ = 3, d₂ = 10
These are the calculator defaults. You have an F statistic of 3.71 from a test with 3 numerator and 10 denominator degrees of freedom, and you want the right-tail probability.
- Confirm the setup: d_1 = 3, d_2 = 10, observed x = 3.71.
- The p-value is the area to the right: P(F \gt 3.71).
- Evaluate the F cumulative distribution at 3.71. The CDF gives P(F \le 3.71) \approx 0.9500.
- Subtract from 1: p = 1 - 0.9500 = 0.0500.
So F = 3.71 sits right at the 5% cutoff. That is not a coincidence: 3.71 is the 0.05 critical value for these degrees of freedom, rounded to two decimals. If you switch to critical mode with \alpha = 0.05, the calculator returns 3.708, which you would reject any observed F above.
Explore how the shape depends on the degrees of freedom
Reading and interpreting the results
The calculator runs in two directions. In p-value mode you supply an observed F and get the right-tail probability. In critical-value mode you supply \alpha and get the cutoff.
- p-value
- The probability of an F at least as large as the one you observed, assuming the null hypothesis is true. Smaller means stronger evidence against the null.
- Critical value
- The F value with exactly \alpha of the area to its right. Reject the null when your observed F exceeds it.
- Significance level \alpha
- The tail area you are willing to treat as rejection territory, chosen before you see the data. Common values are 0.10, 0.05 and 0.01.
The two modes are inverses. In the demo, an observed F of 3.71 gives p = 0.05, and an \alpha of 0.05 gives a critical value of 3.708. When the p-value equals \alpha, the observed F sits exactly on the critical value.
| d₁ | d₂ = 5 | d₂ = 10 | d₂ = 20 | d₂ = 30 |
|---|---|---|---|---|
| 1 | 6.608 | 4.965 | 4.351 | 4.171 |
| 3 | 5.409 | 3.708 | 3.098 | 2.922 |
| 5 | 5.050 | 3.326 | 2.711 | 2.534 |
| 10 | 4.735 | 2.978 | 2.348 | 2.165 |
Common mistakes
The two degrees of freedom are not interchangeable. The 0.05 critical value for d_1 = 3, d_2 = 10 is 3.708, but for d_1 = 10, d_2 = 3 it is 8.786. Swapping them changes the answer by more than double, so always put the numerator df first.
A second error is running a two-tailed test by habit. Most F tests, including ANOVA and the regression overall test, are one-tailed on the right, because large F means the effect variance dominates. Do not double the p-value.
Third, watch the variance-ratio test. When you compare two sample variances, always put the larger variance in the numerator so F \ge 1, then use the appropriate one- or two-sided critical value. If you form the ratio the other way, you land in the left tail where standard tables give no help.
Fourth, remember the normality assumption. The F test for equal variances is famously sensitive to non-normal data. If your data are skewed, a small p-value may reflect the skew, not a true variance difference. In that case a rank-based alternative is safer.
Related calculators
The F distribution is one member of a connected family. For a single mean or slope, use the t distribution calculator, since t^2 with d_2 degrees of freedom equals F with d_1 = 1. For counts and contingency tables, the chi-square calculator handles the numerator and denominator pieces individually. When you need standard scores or areas under a bell curve, the normal distribution calculator and the z-score calculator cover those. The F density itself is a special case of a ratio built from gamma variables, and proportions on a bounded scale often call for the beta distribution calculator.
Frequently asked questions
Why does the F distribution start at zero and not go negative?
Because it is a ratio of two sums of squares, and squares cannot be negative. The smallest possible F is 0, reached only when the numerator variance is exactly 0.
What does an F value close to 1 mean?
It means the two variance estimates are about equal, which is what you expect under the null hypothesis. With d_1 = 3, d_2 = 10 the mean is 1.25 and the median is near 0.85, so a value around 1 is unremarkable and gives a large p-value.
Is the F test one-tailed or two-tailed?
ANOVA and regression F tests are one-tailed on the right. A test comparing two variances can be two-tailed; if so, split \alpha between the tails or use the two-sided form of the test.
How is F related to the t statistic?
Squaring a t statistic with \nu degrees of freedom gives an F statistic with d_1 = 1 and d_2 = \nu. A t of 2.228 with 10 df squares to 4.965, which is the 0.05 F critical value for d_1 = 1, d_2 = 10 in the table above.
What happens as the denominator degrees of freedom grow?
The distribution tightens around 1 and its tail thins. The 0.05 critical value for d_1 = 3 falls from 5.409 at d_2 = 5 to 2.922 at d_2 = 30, so larger samples make the same F look more significant.
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