Brown Forsythe Test: Formula, P Value, Interpretation, R, Python, SPSS and Excel Guide 1

Brown Forsythe Test Overview

The Brown Forsythe Test is mainly used as a robust test of homogeneity of variance. In ordinary one-way ANOVA, researchers often assume that the groups have approximately equal population variances. If this assumption is badly violated, the ordinary ANOVA p-value can become less reliable, especially when the group sizes are unequal.

This post uses student-por.csv, a real student performance dataset. The outcome variable is G3, which represents the final grade. The grouping variable is studytime, a four-level category describing weekly study time. The research question is simple but useful: Does the variance of final grades differ across studytime groups?

The final result is stable across all verified software. R, Python and SPSS all show the same Brown Forsythe Test conclusion: there is no strong evidence that the variance of G3 differs across studytime groups.

What Is the Brown Forsythe Test?

The Brown Forsythe Test is a median-centered version of a Levene-type test. Instead of comparing the raw values directly, the method first converts every observation into its absolute deviation from the group median. Then it runs a one-way ANOVA on those transformed deviation values.

This is why the Brown Forsythe Test is useful in educational research, psychology, business analytics, medical data, social sciences and exam-score analysis. Real datasets are often not perfectly normal. Some groups may contain outliers, skewed values or unequal sample sizes. Brown Forsythe is usually more practical than Bartlett’s test when normality is uncertain.

Brown Forsythe Test Formula

The main Brown Forsythe transformation is:

zᵢⱼ = | yᵢⱼ - median(yⱼ) |

Here, yᵢⱼ is the original value for observation i in group j. The expression median(yⱼ) is the median of group j. The transformed value zᵢⱼ is the absolute distance between the observation and its own group median.

After this transformation, a one-way ANOVA is performed on the transformed values. If the mean absolute deviations are significantly different across groups, the test suggests unequal variances. If they are not significantly different, the test does not find strong evidence of unequal variances.

Brown Forsythe Test Null Hypothesis and Alternative Hypothesis

The hypotheses are easy to remember because the Brown Forsythe Test is about variances, not means.

In this example, the Brown Forsythe Test p-value is 0.3804. Since this value is greater than 0.05, the correct decision is to not reject the equal-variance assumption.

Brown Forsythe vs Levene Test

The Brown Forsythe Test and Levene Test are closely related. The common difference is the center used to measure deviations. A mean-centered Levene Test uses distances from group means. The Brown Forsythe version uses distances from group medians.

In the verified SPSS output, the homogeneity table shows all these versions. The row Based on Median is the Brown Forsythe-style result.

Dataset and Variables Used

This Brown Forsythe Test example uses student-por.csv. The verified Python result file shows 649 rows, 33 columns and 0 missing cells. The main outcome is G3, and the main grouping variable is studytime.

External data source: UCI Machine Learning Repository: Student Performance dataset.

Verified Brown Forsythe Test Results

The Brown Forsythe Test was run in R, Python and SPSS. All three workflows produced the same result.

Group Summary for G3 by Study Time

Brown Forsythe Main Result

P-Value Comparison with Other Variance Tests

The comparison is important. Bartlett gives a significant p-value, but Brown Forsythe, Levene and Fligner-Killeen do not. This is a useful real example showing why Bartlett can be too sensitive when normality is uncertain.

Brown Forsythe Test Result Images

The boxplot shows that final grades generally rise as studytime increases, but this chart alone does not prove that group variances are unequal. It is a visual support chart, while the Brown Forsythe p-value gives the formal variance result.

The p-value comparison chart helps explain the main methodological point. Brown Forsythe is not significant, Levene is not significant and Fligner-Killeen is not significant, while Bartlett is significant. This makes the article stronger than a generic definition page because it shows how different variance tests behave on the same real dataset.

Additional Uploaded Verification Charts

The following additional uploaded charts are included as supporting visual assets. They help document the analysis workflow and give readers more confidence that the result was generated from actual software output rather than only written as theory.

Brown Forsythe Test in R

In R, the Brown Forsythe Test can be calculated manually by converting each G3 value into an absolute deviation from its studytime group median, then running one-way ANOVA on those transformed values.

student <- read.csv("student-por.csv", sep = ";", stringsAsFactors = FALSE)

student$G3 <- as.numeric(student$G3)
student$studytime <- as.factor(student$studytime)

group_medians <- tapply(student$G3, student$studytime, median)
student$bf_abs_dev_G3 <- abs(student$G3 - group_medians[as.character(student$studytime)])

bf_model <- aov(bf_abs_dev_G3 ~ studytime, data = student)
summary(bf_model)

The verified R output gives F(3, 645) = 1.0263 and p = 0.3804.

Brown Forsythe Test in Python

In Python, the median-centered Brown Forsythe version can be run with scipy.stats.levene using center=”median”.

import pandas as pd
from scipy import stats

student = pd.read_csv("student-por.csv", sep=";")

groups = [
    student.loc[student["studytime"] == 1, "G3"],
    student.loc[student["studytime"] == 2, "G3"],
    student.loc[student["studytime"] == 3, "G3"],
    student.loc[student["studytime"] == 4, "G3"]
]

stat, p = stats.levene(*groups, center="median")

print(stat, p)

The verified Python result is statistic = 1.026312 and p = 0.380358, matching the R and SPSS outputs.

Secondary Brown Forsythe Checks from Python

Brown Forsythe Test in SPSS

SPSS can show the Brown Forsythe-style result in the Test of Homogeneity of Variances table. In the SPSS output, the row Based on Median is the important row for the Brown Forsythe version. The verified SPSS output gives Levene Statistic = 1.026, df1 = 3, df2 = 645 and Sig. = 0.380.

SPSS Homogeneity Table Displayed as Text

Manual SPSS Brown Forsythe ANOVA on Absolute Deviations

SPSS Syntax Used

GET DATA
  /TYPE=TXT
  /FILE='D:\low kda score priority basis posts\first post\Brown-Forsythe Test\student_por_spss_clean.csv'
  /ENCODING='UTF8'
  /DELCASE=LINE
  /DELIMITERS=","
  /QUALIFIER='"'
  /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2
  /IMPORTCASE=ALL
  /VARIABLES=
  studytime F1.0
  G3 F2.0
  sex A1
  internet A3
  higher A3
  schoolsup A3
  famsup A3.
CACHE.
EXECUTE.

VALUE LABELS studytime
  1 '<2 hours'
  2 '2 to 5 hours'
  3 '5 to 10 hours'
  4 '>10 hours'.

FREQUENCIES VARIABLES=studytime.

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY
  /MISSING ANALYSIS.

SORT CASES BY studytime.

AGGREGATE
  /OUTFILE=* MODE=ADDVARIABLES
  /BREAK=studytime
  /median_G3_by_studytime=MEDIAN(G3).

COMPUTE bf_abs_dev_G3 = ABS(G3 - median_G3_by_studytime).
EXECUTE.

ONEWAY bf_abs_dev_G3 BY studytime
  /STATISTICS DESCRIPTIVES
  /MISSING ANALYSIS.

Download SPSS verification PDF: Brown Forsythe Test SPSS Output PDF.

Brown Forsythe Test in Excel

Excel does not provide a standard built-in Brown Forsythe button, but the method can be done manually. The key idea is to create absolute deviations from group medians, then run ANOVA on those transformed values.

1. Put studytime in column A.
2. Put G3 in column B.
3. Calculate the median G3 for each studytime group.
4. Calculate each row’s absolute deviation from its group median.
5. Split the transformed deviation values into separate columns by studytime group.
6. Run Data → Data Analysis → ANOVA: Single Factor.
7. The ANOVA F and p-value on those deviation values are the Brown Forsythe result.

=MEDIAN(FILTER($B$2:$B$650,$A$2:$A$650=A2))
=ABS(B2-C2)

For most readers, R, Python or SPSS will be safer than Excel because the manual Excel workflow is more prone to formula-range mistakes.

How to Report the Brown Forsythe Test

A good report should name the test, outcome variable, grouping variable, test statistic, degrees of freedom, p-value and interpretation.

Plain-language report: The spread of final grades was checked across four studytime groups. The Brown Forsythe result was not significant, so the analysis did not show strong evidence that grade variability differs across those groups.

Important: Variance Difference Is Not the Same as Mean Difference

The SPSS output also shows that ordinary one-way ANOVA for mean G3 differences across studytime groups is significant, with F = 15.876 and p < 0.001. This does not contradict the Brown Forsythe result. It means average final grades differ across studytime groups, but the variance test does not find strong evidence that the spread of grades differs.

Common Mistakes in Brown Forsythe Test

1. Treating Brown Forsythe as a normality test

The Brown Forsythe Test is not a normality test. It checks equality of variances across groups.

2. Confusing variance testing with mean testing

Brown Forsythe checks variance. Ordinary ANOVA checks means. These two tests answer different questions.

3. Using Bartlett without considering non-normality

Bartlett can be sensitive to non-normal data. In this example, Bartlett rejects equal variances, while Brown Forsythe, Levene and Fligner-Killeen do not.

4. Reporting only “p is not significant”

A better report includes the test statistic, degrees of freedom, p-value, decision and practical meaning.

5. Saying the variances are exactly equal

A non-significant Brown Forsythe p-value does not prove exact equality. It means the test did not find strong evidence of unequal variances.

Download Brown Forsythe Test Files

The analysis package includes R code, Python code, SPSS syntax, output text files, JSON results, CSV summaries and supporting files used in this guide.

Sources and Method Notes

• UCI Machine Learning Repository: Student Performance dataset
• SciPy documentation: scipy.stats.levene
• RDocumentation: car::leveneTest
• GraphPad Prism Guide: Welch and Brown-Forsythe tests