As an F- test, a group of tests is referred to, in which the test statistic under the null hypothesis follows an F distribution. In the context of regression analysis, a combination of linear ( equation ) is studied hypotheses with the F- test. In the special case of the analysis of variance F- test is a statistical test meant, by means of which it can be decided with a degree of confidence whether two samples from different normally distributed populations differ substantially in terms of their variance. It thus serves among other things, overall review of differences between two statistical populations.

The test goes back to one of the most well-known statistician Ronald Aylmer Fisher ( 1890-1962 ). As a test value of F-test, the F value is calculated, which under the null hypothesis of an F distribution (see also Chi -square distribution ) obeys with and degrees of freedom.

F-test for two samples

The F-test is a term used in mathematical statistics, he refers to a group hypothesis tests with F- distributed test statistic. In the analysis of variance with the F- test of the test meant that from different normally distributed populations analyzed the differences in the variances for two samples.

The F-test assumes that two different normally distributed populations (groups) are given, with the parameters and or and. It is believed that the variance in the second population ( group) may be larger than that in the first population. To check this, a random sample is drawn from each population, where the sample sizes and may also be different. The sampling variables of the first population and the second population have to be independent both within a group and with each other.

For the test of:

Is the F-test, the test statistic is the ratio of the estimated variances of the two samples:

Here are the sample variances and the sample means within the two groups.

Under the null hypothesis, the test statistic is F -distributed with degrees of freedom in the numerator and the denominator. The null hypothesis is rejected for the large values ​​of the test statistic. Man destined to the critical value or to calculate the p- value of the test value. This is most easily done with the aid of an F- value table.

The critical value of K is obtained from the condition:

With the desired level of significance.

The p-value P is calculated by:

With, that found in the sample value of the test statistic.

If you have determined K, then rejects it if. If one calculates the p- value p, rejects you, if.

Frequently 5% is chosen for the significance level of value. However, this is only a common convention, see also the article Statistical significance. However, can be drawn from the obtained probability no direct conclusions on the likelihood of the validity of the alternative hypothesis. If two samples differ already in their variances, they differ widely, of course.


A company wants to change the manufacture of its products to a method that promises better quality. The new method would be more expensive, but should have a smaller scatter. As the test 100 are products prepared by the new process B 120 as compared to products which have been produced with the old method A. The products B have a variance of 80, and the products Tested A is a variance of 95


The test statistic is the test value:

This F value is derived under the null hypothesis of a distribution. The p- value of the sample result is:

Thus, the null hypothesis can not be rejected, and thus the production is not switched to the new method. It remains the question whether this decision is justified. What if the new process is actually a smaller variance causes, but due to the sample, this is undiscovered? But even if the null hypothesis would have been rejected, so a significant difference between the variances would have been found, the difference would have to be negligibly small. First, the question naturally arises as to whether the test would be able to discover the difference. This purpose we consider the test strength. The level of significance is also the minimum value of the test strength. So that leads nowhere. In practice, however, the production would of course only be changed when a considerable improvement would be expected, for example, a decrease in the standard deviation by 25%. How likely is it that the test discovered such a difference? This is exactly the value of the test for strength. The calculation requires first the calculation of the critical value. We assume, and read from a table:


The desired value of the test strength is the chance to discover the mentioned decrease of the standard deviation, ie:

This means that if the variance is decreased by 25% or more, is detected in the at least 91% of the cases.

F-test for multiple sample comparisons

The factorial analysis of variance is also based on the F-test. Here the Treatment and error variances are compared with each other.

F-test of determination of a regression approach

Here it is tested whether the coefficient of determination of the regression approach is zero. If this hypothesis is rejected, one can assume that the chosen regression model has an explanatory value for the regressands Y. For example, it is tested whether several variables together have a significant impact on the regressands. It can thus also happen that the t-test has found the usual significance levels, no significant influence of the individual regressors, the F-test, however, notes the significance of the overall model. The probability that the F- test and the t-test yield different results, increases with the number of degrees of freedom.


  • F- tests are generally examples of the likelihood-ratio test.