Hypothesis Tests

Bartlett's Test

Parametric test for equality of variances of k ≥ 2 normally distributed samples.

Statusvalidated

Version1.0.0

Minitab equivalentStat > ANOVA > Test for Equal Variances (Bartlett)

Description

Bartlett's test examines whether the variances of k independent groups are all equal. It compares the pooled variance s²ₚ to the individual group variances via a χ²-statistic with correction factor C. Bartlett is more powerful than Levene when the data are truly normally distributed — but sensitive to departures from normality. For non-normal data, use Levene-k (Brown-Forsythe) instead.

Formulas

T = \frac{( N - k ) ln ( s _{p}^{2} ) - \sum _{i = 1}^{k} ( n _{i} - 1 ) ln ( s _{i}^{2} )}{C}

Test statistic (χ²-distributed with k−1 degrees of freedom)

C = 1 + \frac{1}{3 ( k - 1 )} (i = 1 \sum k \frac{1}{n _{i} - 1} - \frac{1}{N - k})

Correction factor

s²ₚ

s_{p}^{2} = \frac{\sum _{i = 1}^{k} ( n _{i} - 1 ) s _{i}^{2}}{N - k}

Pooled variance

H₀

H_{0} : σ_{1}^{2} = σ_{2}^{2} = \dots = σ_{k}^{2}

Null hypothesis: All group variances are equal

Assumptions

Observations are independent
Data within each group are normally distributed

Limitations

Very sensitive to departures from normality — prefer Levene-k for non-normal data
χ² approximation inaccurate for very small samples

References

Bartlett, M. S. (1937), Properties of sufficiency and statistical tests, Proc. R. Soc. London A, 160(901), 268–282
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.7