Homogeneity of Variances > Bartlett’s Test

## What is Bartlett’s Test?

There are actually

twotests called Bartlett’s. The first part of this article is forBartlett’s test for homogeneity of variances. If you’re looking forBartlett’s test for sphericity(testing that the correlation matrix has an identity matrix), skip to the bottom section.

**Bartlett’s test for homogeneity of variances** is used to test that variances are equal for all samples. It checks that the assumption of equal variances is true before running certain statistical tests like the One-Way ANOVA. It’s used when you’re fairly certain your data comes from a normal distribution. A similar test, called **Levene’s test**, is a better choice for non normal distributions.

The null hypothesis for the test is that the variances are equal for all samples. In statistics terms, that’s:

H_{0}: σ_{1}^{2} = σ_{2}^{2} = … = σ_{k}^{2}.

The alternate hypothesis (the one you’re testing), is that the variances are not equal for one pair or more:

H_{0}: σ_{1}^{2} ≠ σ_{2}^{2} ≠… ≠ σ_{k}^{2}.

The test statistic is rather ugly:

That’s why you’ll most likely want to use software for the test.

- Instructions for R.
- Excel doesn’t have a built in function, but you can download this Bartlett’s test worksheet and fill in the blanks (Worksheet downloaded from John McDonald at The University of Delaware).
- In STATA, Bartlett’s is added onto ANOVA. The basic syntax of the command is
**oneway dv iv**where*iv*is a categorical variable. For more on running the test in STATA, see*this Notre Dame article*.

If you *really* want to run Bartlett’s test by hand, check out this article, which explains how to do it in great detail.

## Bartlett’s Test for Sphericity

**Bartlett’s test for Sphericity** compares your correlation matrix (a matrix of Pearson correlations) to the identity matrix. In other words, it checks if there is a redundancy between variables that can be summarized with some factors.

- In IBM SPSS 22, you can find the test in the Descriptives menu: Analyse-> Dimension reduction-> Factor-> Descriptives-> KMO and Bartlett’s test of sphericity.
- Instructions in R.

**Reference**:

Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press.

If you prefer an online interactive environment to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

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