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Definition: F-Test

F-test is a statistical test that is used to determine whether two populations having normal distribution have the same variances or standard deviation. This is an important part of Analysis of Variance (ANOVA). However in case the population is non normal, F test may not be used and alternate tests like Bartlett’s test may be used. Generally the comparison of variance is done by comparing the ratio of two variances and in case they are equal the ratio of variances are equal.

In order to carry out the F test we need to first determine the level of significance and then find out the degrees of freedom of numerator and denominator  in order to determine the critical values. The null hypothesis in this case is , H0 : and an appropriate alternate hypothesis is to be used. The F value is calculated as F = . Also the degrees of freedom are n-1 and m-1. This is then compared to the table value of F Statistic for the required confidence interval and degrees of freedom.

For example let us take two methods of measuring particulate matter in water and assume that we want to find out if one is more precise than the other. Precision is measured by the fact that a more precise method shall have a lower standard deviation and hence a lower variance. The data obtained for 10 tests each for the two methods are given below:


Mean (ppm)

Standard Dev










The null hypotheis here is H0 : , and the alternate hypothesis is H1 : .

Now let us calculate F = = 0.49/0.36 = 1.3611. For degrees of freedom of 9 and 95% confidence interval, the Ftab is 3.179.

Therefore F< Ftab and hence the null hypothesis stands accepted and we can say that the standard deviations are not different and hence neither of the methods are precise than the other.


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