# F test

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An **F-test** is any statistical test in which the test statistic has an F-distribution if the null hypothesis is true. A great variety of hypotheses in applied statistics are tested by F-tests. Among these are given below:

- The hypothesis that the means of multiple normally distributed populations, all having the same standard deviation, are equal. This is perhaps the most well-known of hypotheses tested by means of an F-test, and the simplest problem in the analysis of variance.

- The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.

In many cases, the F-test statistic can be calculated through a straightforward process. Two regression models are required, one of which constrains one or more of the regression coefficients according to the null hypothesis. The test statistic is then based on a modified ratio of the sum of squares of residuals of the two models as follows:

Given *n* observations, where model 1 has *k* unrestricted coefficients, and model 0 restricts *m* of the coefficients (typically to zero), the F-test statistic can be calculated as

The resulting test statistic value would then be compared to the corresponding entry on a table of F-test critical values, which is included in most statistical texts.

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