BrownForsytheTest
BrownForsytheTest[data]
tests whether the variance of data is 1.
BrownForsytheTest[{data_{1},data_{2},…}]
tests whether the variances of data_{1}, data_{2}, … are equal.
BrownForsytheTest[dspec,]
tests a dispersion measure against .
BrownForsytheTest[dspec,,"property"]
returns the value of "property".
Details and Options
 BrownForsytheTest tests the null hypothesis against the alternative hypothesis :

data {data_{1},data_{2}} {data_{1},data_{2},…} not all equal  where σ_{i}^{2} is the population variance for data_{i}.
 By default, a probability value or value is returned.
 A small value suggests that it is unlikely that is true.
 The data in dspec must be univariate {x_{1},x_{2},…}.
 The argument can be any positive real number. The default value of is 1 if not specified, and ignored if the number of groups in dspec is more than 2.
 The BrownForsytheTest assumes that the data is normally distributed.
 The BrownForsytheTest is less sensitive to the assumption of normality than the LeveneTest.
 BrownForsytheTest[data,,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
 BrownForsytheTest[data,,"property"] can be used to directly give the value of "property".
 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  When one sample of size is given, the BrownForsytheTest is equivalent to the FisherRatioTest.
 For the sample case, the BrownForsytheTest is a modification of the LeveneTest that replaces the Mean in Abs[data_{ij}Mean[data_{ij}]] with a function . The function fn is generally chosen to be Median, but TrimmedMean[#,1/10]& 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 "TestConclusion" and "ShortTestConclusion" 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 0.05.
 Named settings for VerifyTestAssumptions in BrownForsytheTest include:

"Normality" verify that all data is normally distributed
Examples
open allclose allBasic Examples (2)
Test variances from two populations for equality:
Create a HypothesisTestData object for further property extraction:
Test the ratio of the variances of two populations against a particular value:
Scope (10)
Testing (8)
Test whether the variance of a population is 1:
The value is uniformly distributed in [0,1] under :
The value is typically small when is false:
Compare the variance of a population to a particular value:
Compare the variances of two populations:
The value is uniformly distributed in [0,1] under :
The histogram of a sample of values of the Brown–Forsythe test:
The value is 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 :
Test whether the variances of three populations are identical:
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:
Options (8)
AlternativeHypothesis (3)
SignificanceLevel (2)
VerifyTestAssumptions (3)
Applications (1)
Use the Brown–Forsythe test to determine whether approximate degrees of freedom are needed for a test for equal means:
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 0.05 level, the choice of degrees of freedom affects the test conclusion:
TTest makes the determination to use the Satterthwaite approximation automatically:
Properties & Relations (8)
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[n1] under :
The maximumlikelihood estimate of the degrees of freedom is near :
Given two datasets with lengths and , the test statistic follows an FRatioDistribution[1,n+m2] 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 twosample test statistic:
Typically, the Median is used as a standardizing function:
For data that is found to be heavytailed, the 10% TrimmedMean is used:
LeveneTest is equivalent, but always uses Mean to standardize:
The threesample test statistic:
The Brown–Forsythe test works on the values only when the input is a TimeSeries:
The Brown–Forsythe test works with all the values together when the input is a TemporalData:
Possible Issues (3)
The Brown–Forsythe test assumes the data is drawn from a NormalDistribution:
Use ConoverTest or SiegelTukeyTest for nonnormal data:
The Brown–Forsythe test ignores the argument when there are more than 2 groups:
When there are more than 2 groups in the data, the Brown–Forsythe test only allows the twosided test for the alternative hypothesis: