# Bartlett’s Test (Statistics) – Definition

### Bartlett’s Test (Statistics) Definition

The Bartlett’s Test is a method used in statistics to evaluate whether variances are similar or equal in the available samples. Bartlett’s Test statistics tests for equality of variances across population. Bartlett’s Test was named after a statistician, M. S. Bartlett because of a paper he published in 1937.

This method is used in the comparison of population variances as to whether they are equal or otherwise. For instance, it is commonly assumed that when there are three or more normal population, they have similar variances. The

ANOVA and some DOE analysis results also assume this. If the assumption is however wrong, then the results may be misleading. The Bartlett test is used to verify this assumption.

### A Little More on What is Bartlett’s Test for Homogeneity of Variances

The Bartlett’s Test was introduced when M. S. Bartlett published the paper Properties of Sufficiency and Statistical Tests in 1937. It was named after the founder. The Bartlett’s Test assesses equality in variances drawn from different populations. When comparing three or more populations, the Bartlett’s Test became a method to reckon with.

The Bartlett’s Test has the structure of a hypothesis test. It tests the popular assumption that where there are three or more normal variances, they have the same variance. It checks the validity of this assumption as to where the population variances are equal or otherwise. It uses the null and void alternative hypothesis in carrying out the test.

### Barlett’s Test step by step

Hypotheses

The Bartlett’s Test uses the structure of a hypothesis test, it has step by step measures in testing equality in population variances.

Both null and void alternative hypotheses are used when conducting the tests. Using the null hypothesis, all population variances being tested are compared as equal.

The formula below is applicable;

H0:σ21=σ21=…=σ2k

However, if the population variances being tested are not all equal, the void hypothesis takes its form. That means at least one of the variances differs from the others.

It is essential to know that the Bartlett’s Test does not identify which of the variance is not equal to the others.

### Test statistic

Here are simple steps to follow when calculating equality of population variances using the Bartlett’s Test;

Collect a sample of size (ni from the i-th) from the population.

Identify and calculate the variance from each of the samples.

Estimate the degrees of freedom of the samples. The i-th sample has υi = ni – 1 degrees of freedom, while the overall is υ=∑i=1kυi.

s2=∑i=1kυis2iυ is then realized as the combined sample variance.

However, with regard to Bartlett’s Test, the test statistic must be divided by M, otherwise, it is bias. Hence, the corrected statistics; M/C will be used;

C=1+13(k−1)[(∑i=1k1υi)−1υ].

Critical value

When approximated, M will reflect as χ2k-1 but the approximation is appropriate when ni is at least 5. This birthed a critical value of χ21−α,k−1. (1-α means confidence and k-1 degrees of freedom.

### Conclusion

If the corrected M/C  has a higher value than the critical value, then, one of the population variances differs from the others or not equal to the others. The null hypothesis will be rejected.

The null hypothesis will not be rejected if M/C is less than or equal to the critical value. However, it stipulates that the null hypothesis cannot be rejected due to lack of sufficient evidence. This does not posit that the population variances are all equal, rather, it means that the inequality cannot be proven due to absence of data or insufficiency of proof.

### Academic Research on Bartlett’s Test

Brief report: Bartlett’s test of sphericity and chance findings in factor analysis, Tobias, S., & Carlson, J. E. (1969). Multivariate Behavioral Research, 4(3), 375-377.

Some empirical results concerning the power of Bartlett’s test of the significance of a correlation matrix, Knapp, T. R., & Swoyer, V. H. (1967). American Educational Research Journal, 4(1), 13-17.

On the determination of critical values for Bartlett’s test, Dyer, D. D., & Keating, J. P. (1980). Journal of the American Statistical Association, 75(370), 313-319.

The exact distribution of Bartlett’s test statistic for homogeneity of variances with unequal sample sizes, Chao, M. T., & Glaser, R. E. (1978). Journal of the American Statistical Association, 73(362), 422-426.

Exact critical values for Bartlett’s test for homogeneity of variances, Glaser, R. E. (1976). Journal of the American Statistical Association, 71(354), 488-490.

Exploratory factor analysis: A five-step guide for novices, Williams, B., Onsman, A., & Brown, T. (2010). Australasian Journal of Paramedicine, 8(3).

On Bartlett’s test and Lehmann’s test for homogeneity of variances, Sugiura, N., & Nagao, H. (1969). The Annals of Mathematical Statistics, 2018-2032.

Robust large-sample tests for homogeneity of variances, Layard, M. W. J. (1973). Journal of the American Statistical Association, 68(341), 195-198.

Asymptotic expansions of the distributions of Bartlett’s test and sphericity test under the local alternatives, Nagao, H. (1973). Annals of the Institute of Statistical Mathematics, 25(1), 407-422.

Internationalization of small firms: personal factors revisited, Manolova, T. S., Brush, C. G., Edelman, L. F., & Greene, P. G. (2002). International Small Business Journal, 20(1), 9-31.