statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, Bartlett's test, named after Maurice Stevenson Bartlett, is used to test

homoscedasticity In statistics, a sequence of random variables is homoscedastic () if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as hete ...

, that is, if multiple samples are from populations with equal

variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...

s. Some statistical tests, such as the

analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...

, assume that variances are equal across groups or samples, which can be checked with Bartlett's test. In a Bartlett test, we construct the null and alternative hypothesis. For this purpose several test procedures have been devised. The test procedure due to M.S.E (Mean Square Error/Estimator) Bartlett test is represented here. This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (''k'' − 1) degrees of freedom, where ''k'' is the number of random samples, which may vary in size and are each drawn from independent normal distributions. 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.

Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. This test is used because some common statistical procedures assume that variances of the population ...

and the Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.

Specification

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 sizes

n_i

and

sample variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...

S_i^2

then Bartlett's test statistic is :

\chi^2 = \frac

where

N = \sum_^k n_i

and

S_p^2 = \frac \sum_i (n_i-1)S_i^2

is the pooled estimate for the variance. The test statistic has approximately a

\chi^2_

distribution. Thus, the null hypothesis is rejected if

\chi^2 > \chi^2_

(where

\chi^2_

is the upper tail critical value for the

\chi^2_

distribution). Bartlett's test is a modification of the corresponding

likelihood ratio test In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing ...

designed to make the approximation to the

\chi^2_

distribution better (Bartlett, 1937).

Notes

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

\chi^2 = 2.3026 \frac

References

External links

NIST page on Bartlett's test
{{statistics Analysis of variance Statistical tests

Specification

Notes

See also

References

External links