F-test of Equality of Variances
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A F-test of Equality of Variances is a statistical testing of the null hypothesis that states two normal distributions have equal variances.
- See: F-test, F-distribution, Analysis of variance, Statistical Test, Bartlett's Test, Levene's Test, Brown–Forsythe Test.
References
2016
- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/F-test_of_equality_of_variances Retrieved 2016-08-07
- In statistics, an F-test for the null hypothesis that two normal populations have the same variance is sometimes used, although it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.
- Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the test statistic used is the ratio of two sample variances. This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the F-distribution can be derived. For application in applied statistics, there is concern that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances. In other words, this is a case where "approximate normality" (which in similar contexts would often be justified using the central limit theorem), is not good enough to make the test procedure approximately valid to an acceptable degree.
- The test
- Let X1, ..., Xn and Y1, ..., Ym be independent and identically distributed samples from two populations which each have a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let
- [math]\displaystyle{ \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\text{ and }\overline{Y} = \frac{1}{m}\sum_{i=1}^m Y_i }[/math]
- be the sample means. Let
- [math]\displaystyle{ S_X^2 = \frac{1}{n-1}\sum_{i=1}^n \left(X_i - \overline{X}\right)^2\text{ and }S_Y^2 = \frac{1}{m-1}\sum_{i=1}^m \left(Y_i - \overline{Y}\right)^2 }[/math]
- be the sample variances. Then the test statistic
- [math]\displaystyle{ F = \frac{S_X^2}{S_Y^2} }[/math]
- has an F-distribution with n − 1 and m − 1 degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it has a non-central F-distribution. The null hypothesis is rejected if F is either too large or too small.