F-Test

An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact “F-tests” mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.

Formula and calculation

Most F-tests arise by considering a decomposition of the variability in a collection of data in terms of sums of squares. The test statistic in an F-test is the ratio of two scaled sums of squares reflecting different sources of variability. These sums of squares are constructed so that the statistic tends to be greater when the null hypothesis is not true. In order for the statistic to follow the F-distribution under the null hypothesis, the sums of squares should be statistically independent, and each should follow a scaled χ²-distribution. The latter condition is guaranteed if the data values are independent and normally distributed with a common variance.

Common examples of the use of F-tests include the study of the following cases:

• The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).
• The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of squares.
• The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other.

Assumptions

Several assumptions are made for the test. Your population must be approximately normally distributed (i.e. fit the shape of a bell curve) in order to use the test. Plus, the samples must be independent events. In addition, you’ll want to bear in mind a few important points:

• The larger variance should always go in the numerator (the top number) to force the test into a right-tailed test. Right-tailed tests are easier to calculate.
• For two-tailed tests, divide alpha by 2 before finding the right critical value.
• If you are given standard deviations, they must be squared to get the variances.
• If your degrees of freedom aren’t listed in the F Table, use the larger critical value. This helps to avoid the possibility of Type I errors.

F Test to Compare Two Variances

A Statistical F Test uses an F Statistic to compare two variances, s1 and s2, by dividing them. The result is always a positive number (because variances are always positive). The equation for comparing two variances with the f-test is:

F = S^2 1 / s^2 2

If the variances are equal, the ratio of the variances will equal 1. For example, if you had two data sets with a sample 1 (variance of 10) and a sample 2 (variance of 10), the ratio would be 10/10 = 1.