Free F-Statistic Calculator

What Is an F-Statistic?

An F-statistic (also called an F-ratio) is a test statistic that follows an F‑distribution under the null hypothesis. Its primary purpose is to compare the variances of two normally distributed populations, forming the foundation of the F‑test. The simplest expression of this statistic is a ratio of two sample variances:

$$F = \frac{s_1^2}{s_2^2}$$

where $s_1^2$ and $s_2^2$ are the variances of two independent samples. In practice, the larger variance is placed in the numerator so that $F \geq 1$; this basic version is often referred to as a variance ratio test.

Beyond direct variance comparison, the F‑statistic is a cornerstone of ANOVA (analysis of variance) and linear regression analysis. In regression, the regression F‑statistic tests whether the overall model explains a significant portion of the variance in the dependent variable – i.e., whether at least one explanatory variable has a non‑zero coefficient. This is the central question addressed by an F‑test calculator or F‑distribution calculator when set to “multiple regression” mode.

Performing a Basic F‑Test: A Step‑by‑Step Guide

To conduct a fundamental F‑test for equality of two variances, follow these steps:

1. State the hypotheses.
   The null hypothesis $H_0$ states that the two population variances are equal ($\sigma_1^2 = \sigma_2^2$). The alternative $H_1$ is that they are different ($\sigma_1^2 \neq \sigma_2^2$).

2. Compute the sample variances.
   From each sample, calculate the variances $s_1^2$ and $s_2^2$.

3. Calculate the F‑value.
   Divide the larger variance by the smaller variance: $F = s_{\text{large}}^2 / s_{\text{small}}^2$.

4. Determine the degrees of freedom.
   For a sample of size $n$, the degrees of freedom are $\text{df}_1 = n_1 - 1$ (numerator) and $\text{df}_2 = n_2 - 1$ (denominator).

5. Choose a significance level ($\alpha$).
   Common choices are 0.05 (95% confidence), 0.01, or 0.10.

6. Look up the critical value.
   Using an F‑distribution table or an online variance ratio calculator, find the critical value $F_{\text{crit}}$ for the given $\alpha$, $\text{df}_1$, and $\text{df}_2$.

7. Make a decision.
   If the computed $F > F_{\text{crit}}$, reject $H_0$ and conclude that the variances are significantly different; otherwise, fail to reject $H_0$.

This straightforward procedure requires only the sample data and a reliable F‑table or an F‑test calculator.

The F‑Statistic in Multiple Linear Regression

In multiple regression, the F‑test compares a full (unrestricted) model against a restricted model that omits a subset of predictors. The null hypothesis is that all omitted coefficients are jointly zero – i.e., they do not contribute to explaining the dependent variable.

To compute the F‑statistic, we use the sum of squared residuals (SSR) from both models:

$$F = \frac{(\text{SSR}_{\text{R}} - \text{SSR}_{\text{F}}) / J}{\text{SSR}_{\text{F}} / (N - K)}$$

where:

• $\text{SSR}_{\text{R}}$ = sum of squared residuals from the restricted model,
• $\text{SSR}_{\text{F}}$ = sum of squared residuals from the full model,
• $J$ = number of restrictions (number of coefficients set to zero),
• $K$ = total number of coefficients in the full model (including the intercept),
• $N$ = number of observations.

If the restricted model fits much worse than the full model, $\text{SSR}_{\text{R}}$ will be substantially larger than $\text{SSR}_{\text{F}}$, producing a large F‑value. The F‑statistic then follows an F‑distribution with $J$ and $N-K$ degrees of freedom.

Example
Suppose a regression with $N = 44$, $K = 4$, and we test $J = 2$ restrictions. The critical value at the 5% significance level is approximately $F_{2, 40; 0.05} = 2.606$. If the computed F exceeds this threshold, we reject the null hypothesis and conclude that the set of variables significantly improves the model.

Interpreting the F‑Statistic

A large F‑statistic provides evidence against the null hypothesis. In practice, the result is often reported with a p‑value: if $p < \alpha$, the null is rejected.

Key properties of the F‑distribution:

• It is always positive because variances (and sums of squares) are non‑negative.
• It is right‑skewed (asymmetric); the shape depends on the degrees of freedom.
• As both degrees of freedom increase, the distribution approaches symmetry.

F‑Test vs. T‑Test

While a t‑test assesses the significance of a single coefficient, an F‑test evaluates multiple coefficients simultaneously. For a single restriction, the F‑statistic equals the square of the corresponding t‑statistic. The F‑test is also the core of ANOVA, where it compares between‑group variance to within‑group variance.

Using an F‑Statistic Calculator

An online F‑statistic calculator (often called an F‑test calculator or variance ratio calculator) automates all of the above calculations. You simply input sample variances (or SSR values), degrees of freedom, and a significance level, and the tool returns both the F‑value and its associated p‑value. Such calculators are invaluable for quick hypothesis tests in regression, ANOVA, and any other context that involves comparing variances.