F Statistic Calculator

What Is the F Statistic?

The F statistic compares two independent estimates of variance or tests whether a group of regression coefficients is jointly zero. Under the null hypothesis it follows an F distribution with numerator degrees of freedom $df_1$ and denominator degrees of freedom $df_2$.

Variance Ratio Test

To compare two sample variances $s_1^2$ and $s_2^2$ from independent samples of sizes $n_1$ and $n_2$:

$$F = \frac{s_1^2}{s_2^2}, \quad df_1 = n_1 - 1, \quad df_2 = n_2 - 1$$

A large F suggests the first variance is greater than the second. Use the Variance Calculator to compute sample variances first.

Regression F-Test

When comparing a restricted regression model (SSR$_R$) to a full model (SSR$_F$) with $J$ restrictions, $K$ total coefficients, and $N$ observations:

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

Here $df_1 = J$ and $df_2 = N - K$. A large F indicates the restricted model fits significantly worse, so the omitted variables matter jointly.

How to Use This Calculator

• Choose Variance Ratio or Regression F-Test.
• Enter the required variances, sample sizes, or sum-of-squares values.
• Read the F statistic, degrees of freedom, and upper-tail p-value.
• Compare the p-value to your significance level $\alpha$ (for example 0.05).

Example: Variance Ratio

If $s_1^2 = 10$, $s_2^2 = 5$, $n_1 = 30$, and $n_2 = 30$, then $F = 2$ with $df_1 = 29$ and $df_2 = 29$. The upper-tail p-value tells you how likely such a ratio is when both population variances are equal.

Related Tools

Explore related statistics tools:

• ANOVA Calculator
• Critical Value Calculator
• Degrees of Freedom Calculator