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Computing F-ratio


The F-ratio is used to determine whether the variances in two independent samples are equal. If the F-ratio is not statistically significant, you may assume there is homogeneity of variance and employ the standard t-test for the difference of means. If the F-ratio is statistically significant, use an alternative t-test computation such as the Cochran and Cox method.

Problem:  Given the following summary statistics, are the variances equal?

Sample A =20 n=10

Sample B =30 n=30


Set the Rejection Criteria

Determine the degrees of freedom (df) for each sample

df = n1 - 1 (numerator = n for sample with larger variance)

df for numerator (Sample B) = 29

df = n2 - 1 (denominator = n for sample with smaller variance)

df for denominator (Sample A) = 9

Determine the level of confidence -- alpha

Consult F-Distribution table for df = (29,9), alpha.05

Fcv= 2.70


Compute the Test Statistic


= largest variance

= smallest variance




The test statistic with the f critical value (Fcv) listed in the F distribution. If the F-ratio equals or exceeds the critical value, the null hypothesis (Ho)  (there is no difference between the sample variances) is rejected. If there is a difference in the sample variances, the comparison of two independent means should involve the use of the Cochran and Cox method or one of several alternative techniques.

The test statistic (1.50) did not meet or exceed the critical value (2.70). Therefore, there is no statistically significant difference between the variance exhibited in Sample A and the variance exhibited in Sample B. Assume homogeneity of variance for tests of the difference between sample means.

Software Output Example



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