Bartlett's test

Overview
Bartlett's test (Snedecor and Cochran, 1983) is used to test if k samples have equal variances. Equal variances across samples is called homoscedasticity or homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Bartlett test can be used to verify that assumption.

Bartlett's test is sensitive to departures from normality. That is, if your samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test is an alternative to the Bartlett test that is less sensitive to departures from normality.

Bartlett's test is used to test the null hypothesis, $$H_0$$ that all k population variances are equal against the alternative that at least two are different.

If there are k samples with size $$n_i$$ and sample variance $$S_i^2$$ then Bartlett's test statistic is

$$X^2 = \frac{(N-k)\ln(S_p^2) - \sum_{i=1}^k(n_i - 1)\ln(S_i^2)}{1 + \frac{1}{3(k-1)}\left(\sum_{i=1}^k(\frac{1}{n_i-1}) - \frac{1}{N-k}\right)}$$ where $$N = \sum_{i=1}^k n_i$$ and $$S_p^2 = \frac{1}{N-g} \sum_i (n_i-1)S_i^2$$ is the pooled estimate for the variance.

The test statistic has approximately a $$\chi^2_{k-1}$$ distribution. Thus the null hypothesis is rejected if $$X^2 > \chi^2_{k-1,\alpha}$$ (where $$\chi^2_{k-1,\alpha}$$ is the upper tail critical value for the $$\chi^2_{k-1}$$ distribution).

Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the $$\chi^2_{k-1}$$ distribution better (M. S. Bartlett 1937).