F Distribution Calculator
Find F probabilities below or above a value, the right-tail p-value of an F statistic, or the critical value for a given significance level. Enter the numerator and denominator degrees of freedom; the mean, variance, median and mode come with a shaded density chart.
Positive; the number of groups minus one in a one-way ANOVA
Positive; the total sample size minus the number of groups in a one-way ANOVA
Related Calculators
F-Table (F-Distribution Critical Values)
Look up F critical values by numerator and denominator degrees of freedom for α = 0.10, 0.05, 0.025, and 0.01.
ANOVA Calculator
Run one-way ANOVA calculations and inspect variance across groups.
F-Test for Two Variances
Compare two variances with F, p-value, and df from data or Sx and n — the TI-84 2-SampFTest.
What the F distribution is
An F variable is the ratio of two independent chi-square variables, each divided by its own degrees of freedom. It is positive and skewed to the right, and it is the reference distribution for analysis of variance, for comparing two variances with the F test, and for the overall significance test in regression. Large values of F are evidence against the null hypothesis, so these tests use the right tail.
F = (U₁ / d₁) / (U₂ / d₂), with U₁ ~ χ²(d₁) and U₂ ~ χ²(d₂) independent
Mean = d₂ / (d₂ − 2) for d₂ > 2
Variance = 2 d₂² (d₁ + d₂ − 2) / (d₁ (d₂ − 2)² (d₂ − 4)) for d₂ > 4
Mode = (d₁ − 2) d₂ / (d₁ (d₂ + 2)) for d₁ > 2
If T follows a t distribution with ν degrees of freedom, T² follows F(1, ν)
Which degrees of freedom to use
| Test | Numerator d₁ | Denominator d₂ |
|---|---|---|
| One-way ANOVA with k groups and N observations | k − 1 | N − k |
| Two-way ANOVA effect | factor levels − 1 (interaction: (a − 1)(b − 1)) | error degrees of freedom |
| Ratio of two variances | n₁ − 1 (sample in the numerator) | n₂ − 1 |
| Overall F test in regression with p predictors | p | n − p − 1 |
To turn an F statistic into a p-value choose P(X ≥ x). For the critical value of a one-tailed test at level α choose the upper-tail option with p = α. A two-sided variance-ratio test needs both tails: the upper critical value uses p = α/2 and the lower one uses the lower-tail option with p = α/2.
Critical values
| d₁ | d₂ | 5% in the upper tail | 1% in the upper tail |
|---|---|---|---|
| 1 | 10 | 4.9646 | 10.0443 |
| 2 | 10 | 4.1028 | 7.5594 |
| 3 | 20 | 3.0984 | 4.9382 |
| 4 | 30 | 2.6896 | 4.0179 |
| 5 | 30 | 2.5336 | 3.6990 |
| 2 | 60 | 3.1504 | 4.9774 |
| 10 | 100 | 1.9267 | 2.5033 |
More combinations are in the F table, and the critical value calculator handles z, t, chi-square and F values in one place.
Worked example
A one-way ANOVA with 4 groups and 24 observations has d₁ = 3 and d₂ = 20. The F statistic is significant at the 5% level when it exceeds 3.098391, the value with 5% of the area in the upper tail. Turned around, a statistic of 3.098391 has a p-value of 0.05. Load example shows this critical value with the mean 20/18 = 1.1111, the variance 1.0802, the median 0.8162 and the mode 0.303. The ANOVA calculator computes the F statistic and its p-value from your data.
Software equivalents
| Software | Left tail | Right tail (p-value) | Inverse |
|---|---|---|---|
| Excel / Sheets | F.DIST(x, d1, d2, TRUE) | F.DIST.RT(x, d1, d2) | F.INV(p, d1, d2) and F.INV.RT(α, d1, d2) |
| R | pf(x, d1, d2) | pf(x, d1, d2, lower.tail = FALSE) | qf(p, d1, d2) and qf(α, d1, d2, lower.tail = FALSE) |
| Python (SciPy) | scipy.stats.f.cdf(x, d1, d2) | scipy.stats.f.sf(x, d1, d2) | scipy.stats.f.ppf(p, d1, d2) and scipy.stats.f.isf(α, d1, d2) |
| TI-84 | Fcdf(0, x, d1, d2) | Fcdf(x, 1E99, d1, d2) | no built-in inverse |
Related pages: the Levene test for equal variances and the two-way ANOVA calculator.
Frequently Asked Questions
How do I find a p-value from an F statistic?
Choose P(X ≥ x), enter the F statistic as x and the numerator and denominator degrees of freedom. The result is the area to the right of the statistic, which is the p-value of an ANOVA or variance-ratio F test.
Which degrees of freedom are the numerator and which the denominator?
In a one-way ANOVA the numerator is the between-groups value k − 1 and the denominator is the within-groups value N − k. For a ratio of two variances the numerator belongs to the sample whose variance is on top of the ratio, and each value is that sample size minus one.
How do I find the critical F value?
Choose the option that finds x from an upper-tail probability, enter your significance level α as p, and the two degrees of freedom. For d₁ = 3, d₂ = 20 and α = 0.05 the critical value is 3.098391.
How do I get the lower critical value for a two-sided variance test?
Use the lower-tail option with p = α/2. It equals the reciprocal of the upper α/2 critical value of the F distribution with the two degrees of freedom swapped.
Why does the calculator report an infinite mean or an undefined skewness?
The heavy right tail of an F distribution with few denominator degrees of freedom makes the moments diverge. The mean exists only for d₂ > 2, the variance for d₂ > 4 and the skewness for d₂ > 6.
How is the F distribution related to the t distribution?
The square of a t variable with ν degrees of freedom follows an F distribution with 1 and ν degrees of freedom. For example, the two-sided 5% critical t value 2.228139 with 10 degrees of freedom squares to the F(1, 10) critical value 4.9646.
Embed This Calculator
Add this free calculator to your course page or LMS.
Adjust the height value to fit your page.