Bartlett's test
Encyclopedia
In statistics
, Bartlett's test (Snedecor
and Cochran
, 1983) is used to test if k samples are from populations with equal variance
s. 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 the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test and Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.
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 size
and sample variance 
then Bartlett's test statistic is
where
and 
is the pooled estimate for the variance.
The test statistic has approximately a
distribution. Thus the null hypothesis is rejected if 
(where 
is the upper tail critical value for the 
distribution).
Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the
distribution better (M. S. Bartlett
1937).
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Bartlett's test (Snedecor
George W. Snedecor
George Waddel Snedecor was an American mathematician and statistician. He contributed to the foundations of analysis of variance, data analysis, experimental design, and statistical methodology. Snedecor's F distribution and the George W...
and Cochran
William Gemmell Cochran
William Gemmell Cochran was a prominent statistician; he was born in Scotland but spent most of his life in the United States....
, 1983) is used to test if k samples are from populations with equal variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
s. Equal variances across samples is called homoscedasticity
Homoscedasticity
In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity...
or homogeneity of variances. Some statistical tests, for example the analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...
, 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 the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test and Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.
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 size
and sample variance then Bartlett's test statistic iswhere
and is the pooled estimate for the variance.The test statistic has approximately a
distribution. Thus the null hypothesis is rejected if (where is the upper tail critical value for the distribution).Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the
distribution better (M. S. BartlettM. S. Bartlett
Maurice Stevenson Bartlett FRS was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns...
1937).