In statistics, an Ftest 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 Ftest 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 Fdistribution can be derived.^{[1]} 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.
Contents

The test 1

Properties 2

Generalization 3

See also 4

References 5
The test
Let X_{1}, ..., X_{n} and Y_{1}, ..., Y_{m} 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

\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\text{ and }\overline{Y} = \frac{1}{m}\sum_{i=1}^m Y_i
be the sample means. Let

S_X^2 = \frac{1}{n1}\sum_{i=1}^n \left(X_i  \overline{X}\right)^2\text{ and }S_Y^2 = \frac{1}{m1}\sum_{i=1}^m \left(Y_i  \overline{Y}\right)^2
be the sample variances. Then the test statistic

F = \frac{S_X^2}{S_Y^2}
has an Fdistribution with n − 1 and m − 1 degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it has a noncentral Fdistribution. The null hypothesis is rejected if F is either too large or too small.
Properties
This Ftest is known to be extremely sensitive to nonnormality,^{[2]}^{[3]} so Levene's test, Bartlett's test, or the Brown–Forsythe test are better tests for testing the equality of two variances. (However, all of these tests create experimentwise type I error inflations when conducted as a test of the assumption of homoscedasticity prior to a test of effects.^{[4]}) Ftests for the equality of variances can be used in practice, with care, particularly where a quick check is required, and subject to associated diagnostic checking: practical textbooks^{[5]} suggest both graphical and formal checks of the assumption.
Ftests are used for other statistical tests of hypotheses, such as testing for differences in means in three or more groups, or in factorial layouts. These Ftests are generally not robust when there are violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts.^{[6]} However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the Ftest is relatively robust, although (if the normality assumption does not hold) it suffers from a loss in comparative statistical power as compared with nonparametric counterparts.
Generalization
The immediate generalization of the problem outlined above is to situations where there are more than two groups or populations, and the hypothesis is that all of the variances are equal. This is the problem treated by Hartley's test and Bartlett's test.
See also
References

^ Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0471584940 (Section 27.1)

^

^ Markowski, Carol A; Markowski, Edward P. (1990). "Conditions for the Effectiveness of a Preliminary Test of Variance". The American Statistician 44 (4): 322–326.

^ Sawilowsky, S. (2002). "^{2}_{2} ≠ σ^{2}_{1}"Fermat, Schubert, Einstein, and Behrens–Fisher:The Probable Difference Between Two Means When σ, Journal of Modern Applied Statistical Methods, 1(2), 461–472.

^ Rees, D.G. (2001) Essential Statistics (4th Edition), Chapman & Hall/CRC, ISBN 1584880074. Section 10.15

^ Blair, R. C. (1981). "A reaction to ‘Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance.’" Review of Educational Research, 51, 499–507.
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