# AP Statistics Curriculum 2007 NonParam VarIndep

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

## Contents

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

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*-sample, \[\bar{a} = {1\over N}\sum_{i=1}^{N}{a_{N,i}},\] \[V^2 = {1\over N-1}\sum_{i=1}^{N}{(a_{N,i}-\bar{a})^2}.\]

^{th}**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.

## Alternative tests of Variance Homogeneity

## 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.

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

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