An F statistic is a ratio of two variance estimates. If the null hypothesis is true, both estimate the same quantity and F hovers around 1; a large F says the numerator contains extra variation that the null hypothesis cannot explain. Because both parts are scaled sums of squares, F follows the F distribution with a numerator and a denominator degrees of freedom.
In regression, the F-test checks whether a group of coefficients is jointly zero. You fit a restricted model without those J variables and a full model with K coefficients, then compare their residual sums of squares: if dropping the variables barely worsens the fit, F is small. The overall F-test of a regression is the special case where every slope is dropped, and it can be written using R² alone.
The same statistic appears in one-way ANOVA (between-group versus within-group mean squares) and in the classic test for equal variances of two normal populations, where F is simply the ratio of the sample variances. That test is sensitive to non-normal data, so Levene’s test is often preferred in practice.
How to use the F-Statistic Calculator
- Test type
- Residual sum of squares, restricted model (SSR_R)
- Residual sum of squares, full model (SSR_F)
- Number of excluded coefficients (J)
- Total number of coefficients in the full model (K, incl. intercept)
- Sample size (N)
- Standard deviation of sample 1 (s₁)
- Size of sample 1 (n₁)
- Standard deviation of sample 2 (s₂)
- Size of sample 2 (n₂)
- Coefficient of determination (R²)
- Number of predictors (k)
- Sample size (n)
- Sum of squares between groups (SSB)
- Sum of squares within groups (SSW)
- Number of groups
- Total observations
- Test type
- Significance level (α)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Pick what you know – residual sums of squares, two standard deviations, R² or ANOVA sums of squares – and enter the numbers. The calculator forms the ratio of mean squares, finds the degrees of freedom and reads the p-value and critical value from the F distribution.
Worked example
A wage regression on N = 60 workers has K = 6 coefficients and a residual sum of squares of 150. Dropping J = 3 education variables raises it to 180. F = [(180 − 150) ÷ 3] ÷ [150 ÷ 54] = 10 ÷ 2.778 = 3.60 with (3, 54) degrees of freedom.
The critical F at α = 0.05 is 2.776 and the p-value is 0.019, so the three education variables are jointly significant and should stay in the model.
Where the F-test is used
- Comparing nested regression models or testing groups of dummy variables.
- The overall significance test printed with every regression output.
- One-way and factorial ANOVA.
- Comparing two variances – for example, the precision of two measuring instruments.
Formula
Nested models: F = [(SSR_R − SSR_F) ÷ J] ÷ [SSR_F ÷ (N − K)] · Two variances: F = s₁² ÷ s₂² · From R²: F = (R² ÷ k) ÷ [(1 − R²) ÷ (n − k − 1)] · ANOVA: F = [SSB ÷ (g − 1)] ÷ [SSW ÷ (N − g)]
Worked example
Inputs — Test type: Multiple regression – restricted vs full model; Residual sum of squares, restricted model (SSR_R): 180; Residual sum of squares, full model (SSR_F): 150; Number of excluded coefficients (J): 3; Total number of coefficients in the full model (K, incl. intercept): 6; Sample size (N): 60; Significance level (α): 0.05.
Results — F statistic = 3.6; p-value = 0.02; Decision = Reject H₀ at α = 0.05 – p = 0.0191 ≤ 0.05 and F = 3.6 lies in the rejection region.; Numerator degrees of freedom (df₁) = 3; Denominator degrees of freedom (df₂) = 54; Critical F = 2.7758.
Key terms
- Residual sum of squares (SSR)
- The sum of squared differences between observed values and the model's predictions.
- Restricted model
- The model with some coefficients forced to zero (the null hypothesis).
- Numerator / denominator df
- The two parameters of the F distribution, from the number of restrictions and the residual degrees of freedom.
- Mean square
- A sum of squares divided by its degrees of freedom.
Frequently asked questions
What does a large F statistic mean?
That the numerator variance is much bigger than the denominator variance – for regression, that the excluded variables explain a real share of the variation; for ANOVA, that the group means differ.
What is the critical F for 4 and 40 degrees of freedom?
2.606 at α = 0.05 and 3.828 at α = 0.01.
How are F and t related?
When the numerator has 1 degree of freedom, F = t². Testing a single regression coefficient with an F-test gives the same p-value as its two-tailed t-test.
Can an F statistic be negative?
No. It is a ratio of sums of squares, so it is zero or positive. If SSR_R is smaller than SSR_F, the models are not nested or the numbers are wrong.
Which variance goes on top in the two-variance test?
Either, as long as you use a two-tailed test, which this calculator handles. Many textbooks put the larger variance on top and use a right-tailed table.
How do I get F from R²?
F = (R² ÷ k) ÷ ((1 − R²) ÷ (n − k − 1)). With R² = 0.42, k = 3 and n = 50, F = 11.10 on (3, 46) df, p < 0.0001.