The F Distribution

What is the F distribution used for?

The F distribution is used almost exclusively in hypothesis testing as a distribution of a test statistic under a null hypothesis.

Common hypothesis tests using the F distribution include:

• Two-sample test of variances.
• Analysis of variance (ANOVA).
• Multiple linear regression.

Suppose you are interested in testing whether the variance of two samples drawn from independent, normally distributed populations are equal. You calculate the sample variances s₁² and s₂² from samples of size n₁ and n₂, respectively. Your test statistic is the ratio of these sample variances and has an F distribution with numerator degrees of freedom n₁-1 and denominator degrees of freedom n₂-1. The F distribution is a smooth, continuous distribution characterized by these degrees of freedom.

Several F distributions with different degrees of freedom are shown in the graph below.

How do you use published F distribution tables?

Most people use statistical software, like JMP, to do F tests. Still, many statistics books show F tables, so understanding how to use a table to evaluate your test statistic might be helpful. The steps below describe how to use a typical F table:

1. Identify your alpha level. Typically, there exists a table for popular a values. If you set a = 0.05 for your test, then find the table for a = 0.05 or 1 – a = 0.95.
2. Identify the degrees of freedom for your sample sizes. The rows and columns in an F table correspond to different numerator and denominator degrees of freedom.
3. Find the cell in the table corresponding to your numerator and denominator degrees of freedom. This is the F distribution value.
4. Compare your test statistic to the distribution value and make the appropriate conclusion.