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# Bartlett's test

## for homogeneity of variance

Bartlett's test assesses equality of variances across groups against the alternative hypothesis that variances are unequal for at least two groups. In essence it is comparing the arithmetic mean of your (k) variances, with their geometric mean variance.

The test statistic (T) is calculated as below:

#### Algebraically speaking -

 T  = (N-k)ln(sp2) − Σ(ni - 1) ln(si2) 1 + 1 [Σ( 1 ) − 1 ]   3(k − 1) ni-1 (N − k)
where
• N is the total sample size, Σni
• k is the number of groups,
• ni is the sample size of the ith group,
• si2 is the variance of the ith group.
• sp2is the pooled estimate of variance, (Σ(ni-1)si2)/(N-k)

The value of the test statistic (T) is compared with the upper critical value of the χ2 distribution with k − 1 degrees of freedom at significance level α.

Bartlett's test is sensitive to departures from normality, so should not be used on heavily skewed data. This is sometimes used as a justification to use Levene's test, although Levene's test has its own drawbacks. 