This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BrownForsytheTest

 BrownForsytheTest[data] tests whether the variance of data is 1. BrownForsytheTesttests whether the variances of and are equal. BrownForsytheTesttests a dispersion measure against . BrownForsytheTestreturns the value of .
• BrownForsytheTest performs a hypothesis test on data with null hypothesis that the true population variance , and alternative hypothesis that .
• By default a probability value or -value is returned.
• A small -value suggests that it is unlikely that is true.
• The data must be univariate .
• The argument can be any positive real number.
• Properties related to the reporting of test results include:
 "DegreesOfFreedom" the degrees of freedom used in a test "PValue" list of -values "PValueTable" formatted table of -values "ShortTestConclusion" a short description of the conclusion of a test "TestConclusion" a description of the conclusion of a test "TestData" list of pairs of test statistics and -values "TestDataTable" formatted table of -values and test statistics "TestStatistic" list of test statistics "TestStatisticTable" formatted table of test statistics
• For two samples, the BrownForsytheTest is a modification of the LeveneTest that replaces the mean in Abs[dataij-Mean[datai]] with a function . The function fn is generally chosen to be Median but the TrimmedMean is used if the data is heavy-tailed.
• The following options can be used:
 AlternativeHypothesis "Unequal" the inequality for the alternative hypothesis SignificanceLevel 0.05 cutoff for diagnostics and reporting VerifyTestAssumptions Automatic set which diagnostic tests to run
• For the BrownForsytheTest, a cutoff is chosen such that is rejected only if . The value of used for the and properties is controlled by the SignificanceLevel option. This value is also used in diagnostic tests of assumptions, including tests for normality and symmetry. By default is set to .
 "Normality" verify that all data is normally distributed
Test variances from two populations for equality:
Create a HypothesisTestData object for further property extraction:
Properties of the test:
Test the ratio of the variances of two populations against a particular value:
Perform the test with alternative hypothesis :
Test variances from two populations for equality:
 Out[2]=
Create a HypothesisTestData object for further property extraction:
 Out[3]=
Properties of the test:
 Out[4]=

Test the ratio of the variances of two populations against a particular value:
 Out[2]=
 Out[3]=
Perform the test with alternative hypothesis :
 Out[4]=
 Scope   (9)
Test whether the variance of a population is 1:
The -values are typically large under :
The -values are typically small when is false:
Compare the variance of a population to a particular value:
Compare the variances of two populations:
The -values are typically large when the variances are equal:
The -values are typically small when the variances are not equal:
Test whether the ratio of the variances of two populations is a particular value:
The following forms are equivalent:
The order of the datasets should be considered when determining :
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Extract some properties from a HypothesisTestData object:
The -value, test statistic, and degrees of freedom:
Extract any number of properties simultaneously:
The -value, test statistic, and degrees of freedom:
Tabulate test results:
The values from the table can be extracted using :
Tabulate -values or test statistics:
The -value from the table:
The test statistic from the table:
 Options   (8)
A two-sided test is performed by default:
Test versus :
Perform a two-sided test or a one-sided alternative:
Test versus :
Test versus :
Test versus :
Perform tests with one-sided alternatives when a null value is given:
Test versus :
Test versus :
Set the significance level for diagnostic tests:
By default, is used:
The significance level is also used for and :
Diagnostics can be controlled as a group using All or None:
Verify all assumptions:
Check no assumptions:
Diagnostics can be controlled independently:
Check for normality:
It is often useful to bypass diagnostic tests for simulation purposes:
The assumptions of the test hold by design, so a great deal of time can be saved:
The results are identical:
 Applications   (1)
Use the Brown-Forsythe test to determine whether approximate degrees of freedom are needed for a -test for equal means:
A two-sample -test:
If the two samples have equal variances the following degrees of freedom can be used; otherwise, a Satterthwaite approximation is needed:
The Brown-Forsythe test suggests the variances are not equal:
At the level the choice of degrees of freedom affects the test conclusion:
TTest makes the determination to use the Satterthwaite approximation automatically:
The Brown-Forsythe test is equivalent to FisherRatioTest when a single dataset is given:
Given a single dataset with length , the test statistic follows a ChiSquareDistribution under :
The maximum-likelihood estimate of the degrees of freedom is near :
Given two datasets with lengths and , the test statistic follows an FRatioDistribution under :
The Brown-Forsythe test is less sensitive to the assumption of normality than the FisherRatioTest given two datasets:
The Fisher ratio test tends to underestimate the -value and commit more Type I errors:
The two-sample test statistic:
Typically, the Median is used as a standardizing function:
For data that is found to be heavy tailed, the 10% TrimmedMean is used:
LeveneTest is equivalent but always uses Mean to standardize:
The Brown-Forsythe test assumes the data is drawn from a NormalDistribution:
Use ConoverTest or SiegelTukeyTest for non-normal data:
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