## Quick Reference

A test for equality of two or more variances. Suppose the sample variance estimate for the *j*th sample is *s*_{j}^{2} based on *ν** _{j}* degrees of freedom. In the case of samples from two populations having normal distributions, the F-test compares the ratio

*s*_{1}^{2}/*s*_{2}^{2}

with the critical values of an *F*-distribution with *ν*_{1} and *ν*_{2} degrees of freedom. The *F*-test is encountered most frequently in the context of an analysis of variance table (see ANOVA), where it is often referred to as a variance-ratio test.

To test the hypothesis that *m* normal populations have the same variance, the Bartlett test (suggested by Bartlett in 1937) has test statistic, *B*, defined by: where *ν*=*ν*_{1}+*ν*_{2}+…+*ν** _{m}* and . If the null hypothesis of equal variance is correct, then

*B*has a chi-squared distribution with (

*m*− 1) degrees of freedom.

Alternatives in cases where the sample sizes are equal are the Cochran *C* test, introduced by Cochran in 1941, and the Hartley test introduced by Hartley in 1950. The Cochran test statistic, *C*, is given by and the Hartley test statistic, *H*, is given by

*H*=*s*^{2}_{max}/*s*^{2}_{min},

where *s*^{2}_{max} and *s*^{2}_{min} are, respectively, the maximum and the minimum of *s*^{2}_{1}, *s*^{2}_{2},…, *s*^{2}* _{m}*. Unusually high values of

*C*or

*H*indicate unequal variances.

All of these tests are sensitive to departures from normality. See also test for equality of scale.

*Subjects:*
Probability and Statistics.

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