# F-Statistic Calculator

URL: https://speedycalculator.com/calculators/statistics/f-statistic/
Last updated: 2026-10-30

> The F-statistic calculator computes F, its degrees of freedom, the p-value and the critical F for nested regression models (restricted vs full), the ratio of two sample variances, the overall significance of a regression from R², or a one-way ANOVA from sums of squares.

## Inputs

- 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)

## Results

- F statistic
- p-value
- Decision
- Numerator degrees of freedom (df₁)
- Denominator degrees of freedom (df₂)
- Critical F
- F distribution with rejection region

## 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)]
```

## Explanation

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.

## 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.

## FAQ

**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.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [Penn State STAT 462 – Applied Regression Analysis](https://online.stat.psu.edu/stat462/)
- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)