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. Equal variances across samples is called homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples.

Approach

The (modified) Fligner-Killeen test for homogeneity of variances of k populations jointly ranks the absolute values $|X_{i,j}-\tilde{X_j}|$ and assigns increasing scores $a_{N,i}=\Phi^{-1}({1 + {i\over N+1} \over 2})$, 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 jth population, and \Phi(.) is the cummulative distribution function for Normal distirbution. The Fligner-Killeen test is sometimes also called the median-centering Fligner-Killeen test.

• Fligner-Killeen test statistics:
Failed to parse (syntax error): 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 jth sample, a is the overall mean score of all aN,i, and V2 is the sample variance of all scores.

That is:

$N=\sum_{j=1}^k{n_j}$,
$\bar{A_j} = {1\over n_j}\sum_{i=1}^{n_j}{a_{N,i}}$,
$\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}}$,
$V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}$.
• 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.

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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.