Bartlett's test

Bartlett's test is used to test the null hypothesis, H₀ that all k population variances are equal against the alternative that at least two are different.

If there are k samples with sizes ${\displaystyle n_{i}}$ and sample variances ${\displaystyle S_{i}^{2}}$ then Bartlett's test statistic is

${\displaystyle \chi ^{2}={\frac {(N-k)\ln(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{i=1}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}}$

where ${\displaystyle N=\sum _{i=1}^{k}n_{i}}$ and ${\displaystyle S_{p}^{2}={\frac {1}{N-k}}\sum _{i}(n_{i}-1)S_{i}^{2}}$ is the pooled estimate for the variance.

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

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

The test statistics may be written in some sources with logarithms of base 10 as:[2]

${\displaystyle \chi ^{2}=2.3026{\frac {(N-k)\log _{10}(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\log _{10}(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{i=1}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}}$

• Box's M test
• Levene's test

• Bartlett, M. S. (1937). "Properties of sufficiency and statistical tests". Proceedings of the Royal Statistical Society, Series A 160, 268–282 JSTOR 96803
• Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press. ISBN 978-0-8138-1561-9

• NIST page on Bartlett's test