AP Statistics Curriculum 2007 NonParam VarIndep

General Advance-Placement (AP) Statistics Curriculum - Variances of Two Independent Samples

Differences of Variances of Independent Samples

It is frequently necessary to test if k samples have equal variances. Homogeneity of variances is often a reference to equal variances across samples. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples.

Approach

The (modified) Fligner-Killeen test provides the means for studying the homogeneity of variances of k populations { \(X_{i,j}\), for \(1\leq i \leq n_j\) and \(1\leq j \leq k\)}. The test jointly ranks the absolute values of \(|X_{i,j}-\tilde{X_j}|\) and assigns increasing scores \(a_{N,i}=\Phi^{-1}\bigg ({1 + {i\over N+1} \over 2} \bigg )\), based on the ranks of all observations, see the Conover, Johnson, and Johnson (1981) reference below.

In this test, \(\tilde{X_j}\) is the sample median of the j^th population, and \(\Phi(.)\) is the cumulative distribution function for Normal distribution. The Fligner-Killeen test is sometimes also called the median-centering Fligner-Killeen test.

• Fligner-Killeen test statistics:

\[x_o^2 = {\sum_{j=1}^k {n_j(\bar{A_j} -\bar{a})^2} \over V^2},\]

where \(\bar{A_j}\) is the mean score for the j^th sample, \(\bar{a}\) is the overall mean score of all \(a_{N,i}\), and \(V^2\) is the sample variance of all scores.

That is: \[N=\sum_{j=1}^k{n_j},\] \[\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,m_i}},\] where \(a_{N,m_i}\) is the increasing rank score for the i^th-observation in the j^th-sample, \[\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}},\] \[V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}.\]

• Fligner-Killeen probabilities:

For large sample sizes, the modified Fligner-Killeen test statistic has an asymptotic chi-square distribution with (k-1) degrees of freedom \[x_o^2 \sim \chi_{(k-1)}^2.\]

• Note:

Conover, Johnson, and Johnson (1981) carried a simulation comparing different variance homogeneity tests and reported that the modified Fligner-Killeen test is most robust against departures from normality.

Computational Resources: Internet-based SOCR Tools

• See the SOCR Fligner-Killeen Analysis applet.
• See the SOCR Fligner-Killeen Activity.

Examples

Suppose we wanted to study whether the variances in certain time period (e.g., 1981 to 2006) of the consumer-price-indices (CPI) of several items were significantly different. We can use the SOCR CPI Dataset to answer this question for the Fuel, Oil, Bananas, Tomatoes, Orange Juice, Beef and Gasoline items.

See also

• SOCR Fligner-Killeen Activity provides more hands-on examples.
• Parametric Variance Homogeneity test.
• SOCR Fligner-Killeen Applet.

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Alternative tests of Variance Homogeneity

• Levene Test
• Bartlett Test
• Brown-Forsythe test

References

• Conover, W. J., Johnson, M.E., and Johnson M. M. (1981), A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23, 351-361.

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• SOCR Home page: http://www.socr.ucla.edu

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