0.157791
1.111111
0.709877
0.545455
0.842542
0.758288
0.157791
1.111111
0.709877
0.545455
0.842542
0.758288
The F-Distribution Calculator computes the probability density function (PDF) and key statistics of the F-distribution (also called the Fisher-Snedecor distribution), a fundamental distribution in analysis of variance (ANOVA), regression analysis, and comparing variances. The F-distribution is defined as the ratio of two independent chi-square random variables, each divided by their respective degrees of freedom. It is parameterized by two degrees of freedom: d₁ (numerator) and d₂ (denominator).
The F-distribution is the workhorse of ANOVA (Analysis of Variance), the statistical method for comparing means of three or more groups. The F-statistic in ANOVA is the ratio of between-group variance to within-group variance; under the null hypothesis that all group means are equal, this ratio follows an F-distribution. Similarly, in multiple regression, the overall F-test evaluates whether any predictor in the model is significantly related to the response variable. The F-distribution also underpins Levene's test and Bartlett's test for equality of variances across groups.
The F-distribution is defined only for positive values (x > 0) and is right-skewed, especially for small degrees of freedom. The mean is d₂/(d₂ − 2) for d₂ > 2, which is close to 1 for large d₂, reflecting the expectation that the variance ratio is approximately 1 under the null hypothesis. The mode is always less than 1 when d₁ > 2, showing that the most probable F-value is slightly below the mean. As both degrees of freedom increase, the F-distribution becomes more concentrated around 1.
The relationship between the F-distribution and other distributions is fundamental: the square of a t-distributed variable with ν degrees of freedom follows an F(1, ν) distribution, and if X ~ F(d₁, d₂), then 1/X ~ F(d₂, d₁). This reciprocal property means that testing whether one variance is larger than another at significance level α is equivalent to testing whether the reciprocal is smaller at the same level. The F-distribution's mean exists only for d₂ > 2 and its variance only for d₂ > 4, reflecting the heavy-tailed nature for small denominator degrees of freedom.
The F-distribution PDF with d₁ and d₂ degrees of freedom is:
$$f(x; d_1, d_2) = \frac{1}{x \cdot B(d_1/2, d_2/2)} \left(\frac{d_1}{d_2}\right)^{d_1/2} \frac{x^{d_1/2 - 1}}{\left(1 + \frac{d_1 x}{d_2}\right)^{(d_1 + d_2)/2}}$$
Equivalently, using the gamma function:
$$\ln f = \ln\Gamma\!\left(\frac{d_1+d_2}{2}\right) - \ln\Gamma\!\left(\frac{d_1}{2}\right) - \ln\Gamma\!\left(\frac{d_2}{2}\right) + \frac{d_1}{2}\ln\!\left(\frac{d_1}{d_2}\right) + \left(\frac{d_1}{2}-1\right)\ln x - \frac{d_1+d_2}{2}\ln\!\left(1 + \frac{d_1 x}{d_2}\right)$$
The key moments are:
$$\mu = \frac{d_2}{d_2 - 2}\;(d_2 > 2), \quad \text{Mode} = \frac{d_1 - 2}{d_1} \cdot \frac{d_2}{d_2 + 2}\;(d_1 > 2)$$
The PDF value indicates the relative likelihood of observing a particular F-value. In ANOVA and regression, large F-values (in the right tail) provide evidence against the null hypothesis. The mean near 1.0 (for large d₂) reflects the expected ratio under the null hypothesis. The mode is the most probable F-value, always slightly below the mean. The variance decreases as both df increase, meaning more data produces a more concentrated sampling distribution and more powerful tests. When interpreting an F-test, an observed statistic much larger than the mean suggests significant effects.
Inputs
Results
F(3,36)=3.2 in a one-way ANOVA comparing 4 groups (df1=3) with 40 total observations (df2=36). The critical value at α=0.05 is 2.87, so this result (3.2 > 2.87) is significant.
Inputs
Results
F(5,50)=2.4 tests whether 5 predictors jointly explain significant variance. The critical value at α=0.05 is approximately 2.40, making this a borderline significant result.
In ANOVA, the F-statistic is the ratio of between-group variance (MSB) to within-group variance (MSW). Under the null hypothesis that all group means are equal, this ratio follows an F(k−1, N−k) distribution, where k is the number of groups and N is the total sample size. A large F-value indicates that the between-group variability is much larger than would be expected by chance, providing evidence that at least one group mean differs from the others.
The numerator df (d₁) represents the number of independent comparisons or parameters being tested. In one-way ANOVA, d₁ = k − 1 (number of groups minus 1). In regression, d₁ = number of predictors. The denominator df (d₂) represents the residual degrees of freedom, reflecting the amount of data available to estimate the error variance. Larger d₂ gives more precise variance estimates and more powerful tests.
If T follows a t-distribution with ν degrees of freedom, then T² follows an F(1, ν) distribution. This means a two-sample t-test comparing two groups is equivalent to an F-test (ANOVA) with 2 groups. The F-test generalizes the t-test to handle more than two groups simultaneously. For comparing exactly two means, both approaches give identical p-values: p(F) = p(two-tailed t).
The mean d₂/(d₂ − 2) is only defined for d₂ > 2. For d₂ ≤ 2, the right tail of the distribution is so heavy that the expected value is infinite. Similarly, the variance exists only for d₂ > 4. In practice, this means that F-tests with very small denominator degrees of freedom (e.g., comparing variances with only 2-3 observations per group) have poor statistical properties and wide confidence intervals.
In multiple regression, the overall F-test evaluates H₀: all regression coefficients are zero (the model has no predictive power). The F-statistic is (explained variance / df_model) / (residual variance / df_residual), following an F(p, n−p−1) distribution where p is the number of predictors. Additionally, partial F-tests compare nested models to determine whether adding variables significantly improves fit. The R² change test is also based on the F-distribution.
If X ~ F(d₁, d₂), then 1/X ~ F(d₂, d₁). This reciprocal property is practically useful: testing whether variance A exceeds variance B is equivalent to testing whether variance B is less than variance A with swapped degrees of freedom. It also means you only need upper-tail critical values — lower-tail values can be obtained by taking the reciprocal of the upper-tail value with swapped df. For example, F_{0.05}(5, 10) = 1/F_{0.95}(10, 5).
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