Bartlett's Test for Homogeneity of Variances
Also known as: Bartlett's Chi-Square Test, Test for Equality of Variances, Bartlett's Homogeneity Test, Varyans Homojenliği Testi
Bartlett's Test is a classical parametric procedure for testing whether two or more independent groups share a common population variance. Introduced by Maurice Stevenson Bartlett in 1937, it formalises the null hypothesis that all group variances are equal by constructing a chi-square statistic from the ratio of pooled to individual group variances. It is a standard pre-analysis step before applying ANOVA or other procedures whose validity depends on the homoscedasticity assumption.
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When to use it
Bartlett's Test is appropriate when comparing variances across two or more independent, normally distributed groups before conducting ANOVA or similar analyses. It assumes that each group's observations are independently and normally distributed; the test is notably sensitive to departures from normality, which can inflate Type I error. When normality is uncertain, Levene's or Fligner-Killeen tests are recommended alternatives. Best suited to continuous data with reasonably large within-group sample sizes.
Strengths & limitations
- Optimal power under normality: when the normality assumption holds, Bartlett's test is asymptotically the most powerful test for variance equality.
- Chi-square reference distribution: the analytic chi-square approximation avoids resampling, making the test fast and easy to implement.
- Handles multiple groups: extends naturally to k groups simultaneously, unlike simple two-sample F-tests.
- Widely supported: available in virtually all major statistical software packages and well-established in the literature.
- Sensitive to non-normality: even mild departures from normality can substantially inflate the Type I error rate, making the test unreliable without prior normality verification.
- Poor performance with small samples: the chi-square approximation deteriorates when group sample sizes are small, leading to inaccurate p-values.
- Does not identify which groups differ: a significant result indicates at least one group variance differs, but post-hoc procedures are needed to localise the difference.
- Restricted to continuous data: the test is not appropriate for ordinal or categorical variables.
Frequently asked
How does Bartlett's test differ from Levene's test?
Bartlett's test is based on likelihood-ratio principles and assumes normality, giving it optimal power under that assumption but making it sensitive to non-normal data. Levene's test uses absolute deviations from group means or medians and is far more robust to non-normality, making it the preferred choice when the distribution of residuals is uncertain.
What should I do if Bartlett's test is significant?
A significant result means the equal-variance assumption is likely violated. Depending on context, you may apply variance-stabilising transformations, switch to Welch's ANOVA (which does not require homoscedasticity), or use non-parametric alternatives. Post-hoc inspection of group standard deviations helps identify which groups drive the heterogeneity.
Can Bartlett's test be used with only two groups?
Yes. With two groups, Bartlett's test is equivalent to the two-sample F-test for equality of variances. However, even for two groups the test retains its sensitivity to non-normality, so Levene's or Fligner-Killeen tests remain better choices whenever the normality of each group cannot be confirmed.
Sources
- Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London. Series A, 160(901), 268–282. DOI: 10.1098/rspa.1937.0109 ↗
How to cite this page
ScholarGate. (2026, June 2). Bartlett's Test for Homogeneity of Variances. ScholarGate. https://scholargate.app/en/statistics/bartlett-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Fligner-Killeen TestStatistics↔ compare
- One-way ANOVAStatistics↔ compare