Hartley F Max Test: Formula, Assumptions, Excel, R, Python, SPSS and Real Data Example 1

What Is the Hartley F Max Test?

The Hartley F Max Test, also written as the Hartley F-Max test, Hartley test, or F max test, is used to check whether several groups have approximately equal variances. It is commonly discussed before ANOVA because ordinary ANOVA assumes that the variability within groups is not severely different.

The idea is simple. First, calculate the sample variance for every group. Second, find the largest variance and the smallest variance. Third, divide the largest variance by the smallest variance. If the ratio is close to 1, group variances are similar. If the ratio is very large, one group is much more variable than another, and the equal variance assumption may be doubtful.

If you are learning variance-testing workflows, you may also find these related guides useful: Cochran C Test for checking the largest variance relative to total variance, Brown Forsythe Test for robust variance comparison, Goldfeld Quandt Test for regression heteroscedasticity, and Greenhouse Geisser Correction for repeated-measures sphericity correction.

Hartley F Max Test Formula

The Hartley F Max statistic is calculated from group variances. If the group sample variances are written as s₁², s₂², s₃² and so on, the formula is:

Fmax = largest group variance / smallest group variance

For the study-time example, the largest variance is 10.5179 and the smallest variance is 6.2605. Therefore:

Fmax = 10.5179 / 6.2605 = 1.6800

The statistic is always at least 1 because the largest variance cannot be smaller than the smallest variance. A value close to 1 means that the group variances are very similar. A larger value means that the most variable group is much more spread out than the least variable group.

Rule-of-thumb interpretation

There is no single universal F-Max cutoff that works for every dataset. The formal decision depends on the number of groups, group sizes, degrees of freedom and significance level. As a practical screening rule, a modest F-Max value such as 1.6800 usually does not suggest a severe variance problem, but the final conclusion should still consider assumptions and robust tests.

Critical-value interpretation

The classical Hartley test compares the observed F-Max statistic with a critical value from Hartley F-Max tables. These tables are designed mainly for equal sample sizes and normal populations. If the observed F-Max is larger than the critical value, the equal variance assumption is rejected. If the observed F-Max is smaller than the critical value, the test does not reject homogeneity of variance.

Hartley F Max Test Null Hypothesis and Alternative Hypothesis

Dataset and Grouping Variable Used

This worked example uses the student-por.csv student performance dataset. The main outcome variable is G3, the final grade. The main grouping variable is studytime, which divides students into four study-time categories.

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

Verified Results in R, Python and SPSS

The analysis was reproduced through R, Python and SPSS workflows. R and Python were used to calculate group variances, F-Max values, permutation evidence and charts. SPSS was used to verify descriptive statistics, Levene-style homogeneity checks, group sizes and the final Hartley F-Max calculation.

Main Hartley F Max Result for G3 by Study Time

Hartley F Max Calculation Summary

SPSS Levene Test Comparison

F-Max Comparison for G1, G2 and G3

Hartley F Max Test Charts and Interpretation

1. Hartley F-Max Statistic Across Grouping Variables

This chart compares the Hartley F-Max statistic across several possible grouping variables. The highest descriptive variance ratio appears for Medu, followed by traveltime and school. This does not automatically prove unequal variance, but it shows where variance differences are largest and where a researcher may want to inspect group distributions more carefully.

2. Permutation Null Distribution for Hartley F-Max

The permutation distribution is helpful because the actual study-time groups are unbalanced. Instead of relying only on a classical equal-size Hartley table, permutation logic reshuffles group labels many times and asks whether the observed F-Max is unusually large compared with random group assignments. This gives a more realistic diagnostic for applied data.

3. Hartley F-Max Statistic for G1, G2 and G3

The grade comparison shows that G3 has the largest F-Max statistic among the three grade variables. This means final grades have the strongest variance contrast across study-time groups, although the ratio is still not extreme.

4. G3 Distribution by Study Time Group

The boxplot explains why the first two groups have higher variance. The lower study-time groups include very low G3 values, including zero-grade observations, which increase spread. The 5 to 10 hours group has a higher minimum and a narrower spread, which explains why it has the smallest variance.

5. Hartley F Max Test Group Variances for G3

This chart is the visual version of the Hartley formula. The tallest bar is the largest variance, and the shortest bar is the smallest variance. Dividing the tallest by the shortest gives the F-Max statistic.

6. Mean G3 with Standard Error by Study Time Group

This chart is not the Hartley test itself, but it helps explain the data. Mean G3 generally increases from the lowest study-time group to the 5 to 10 hours group. The >10 hours group has a high mean too, but it has a much smaller sample size, so its standard error is larger.

7. Hartley F Max Test P-Value Comparison

The p-value comparison gives a stronger applied view than the raw F-Max chart alone. A small permutation p-value means the observed variance ratio is unusually large under random reshuffling. Variables above the 0.05 threshold do not show strong permutation evidence of unequal variance.

8. Sample Size by Study Time Group

This chart is essential for interpreting the Hartley result. The largest group has 305 students, while the smallest group has only 35 students. Because Hartley’s classical form works best with equal or nearly equal sample sizes, this imbalance is a major reason to interpret the result cautiously.

9. G3 Variance Versus Mean by Study Time Group

This chart shows that the 5 to 10 hours group has one of the highest mean scores and the lowest variance. The first two study-time groups have lower mean scores and higher spread, mainly because they contain more low-grade observations.

How to Run the Hartley F Max Test in R, Python, SPSS and Excel

Hartley F Max Test in R

In R, you can calculate the Hartley F-Max statistic by grouping the outcome variable, calculating sample variances, and dividing the largest variance by the smallest variance. A permutation benchmark can also be added for applied interpretation.

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

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

dat <- student[!is.na(student$G3) & !is.na(student$studytime), ]

group_summary <- aggregate(G3 ~ studytime, data = dat, FUN = function(x) {
  c(n = length(x), mean = mean(x), sd = sd(x), variance = var(x))
})

group_summary_clean <- data.frame(
  studytime = group_summary$studytime,
  n = group_summary$G3[, "n"],
  mean = group_summary$G3[, "mean"],
  sd = group_summary$G3[, "sd"],
  variance = group_summary$G3[, "variance"]
)

print(group_summary_clean)

largest_variance <- max(group_summary_clean$variance)
smallest_variance <- min(group_summary_clean$variance)
hartley_fmax <- largest_variance / smallest_variance

cat("Largest variance:", largest_variance, "\n")
cat("Smallest variance:", smallest_variance, "\n")
cat("Hartley F-Max:", hartley_fmax, "\n")

set.seed(123)
B <- 20000
sim_fmax <- numeric(B)

for(i in 1:B){
  shuffled_group <- sample(dat$studytime)
  temp <- aggregate(dat$G3, by = list(group = shuffled_group), FUN = var)
  sim_fmax[i] <- max(temp$x) / min(temp$x)
}

perm_p <- mean(sim_fmax >= hartley_fmax)
crit_95 <- quantile(sim_fmax, 0.95)

cat("Permutation p-value:", perm_p, "\n")
cat("Permutation 95% critical value:", crit_95, "\n")

Hartley F Max Test in Python

In Python, use pandas to calculate group variances and NumPy to create a permutation benchmark. This method is reproducible and works well for automated post workflows.

import numpy as np
import pandas as pd

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

student["G3"] = pd.to_numeric(student["G3"], errors="coerce")
student["studytime"] = pd.to_numeric(student["studytime"], errors="coerce")

dat = student[["G3", "studytime"]].dropna().copy()

group_summary = (
    dat.groupby("studytime")["G3"]
    .agg(n="count", mean="mean", std="std", variance="var")
    .reset_index()
)

print(group_summary)

largest_variance = group_summary["variance"].max()
smallest_variance = group_summary["variance"].min()
hartley_fmax = largest_variance / smallest_variance

print("Largest variance:", round(largest_variance, 4))
print("Smallest variance:", round(smallest_variance, 4))
print("Hartley F-Max:", round(hartley_fmax, 4))

rng = np.random.default_rng(123)
B = 20000
sim_fmax = np.empty(B)

y = dat["G3"].to_numpy()
g = dat["studytime"].to_numpy()

for i in range(B):
    shuffled = rng.permutation(g)
    temp = pd.DataFrame({"G3": y, "group": shuffled})
    variances = temp.groupby("group")["G3"].var()
    sim_fmax[i] = variances.max() / variances.min()

perm_p = np.mean(sim_fmax >= hartley_fmax)
crit_95 = np.quantile(sim_fmax, 0.95)

print("Permutation p-value:", round(perm_p, 4))
print("Permutation 95% critical value:", round(crit_95, 4))

Hartley F Max Test in SPSS

SPSS does not have a direct menu button named Hartley F Max Test. The statistic can be calculated from group standard deviations and variances. The workflow below also runs Levene’s test for supporting evidence.

* Hartley F Max Test in SPSS: G3 by studytime.

DATASET ACTIVATE HartleyData.

MEANS TABLES=G3 BY studytime
 /CELLS=COUNT MEAN MEDIAN STDDEV VARIANCE MIN MAX SEMEAN.

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

DATASET COPY HartleyMainG3.
DATASET ACTIVATE HartleyMainG3.

AGGREGATE
 /OUTFILE=*
 /BREAK=studytime
 /group_n=N
 /g3_mean=MEAN(G3)
 /g3_median=MEDIAN(G3)
 /g3_sd=SD(G3)
 /g3_min=MIN(G3)
 /g3_max=MAX(G3).

COMPUTE g3_variance = g3_sd ** 2.
EXECUTE.

AGGREGATE
 /OUTFILE=* MODE=ADDVARIABLES
 /BREAK=
 /max_variance=MAX(g3_variance)
 /min_variance=MIN(g3_variance)
 /min_group_n=MIN(group_n)
 /max_group_n=MAX(group_n).

COMPUTE hartley_fmax = max_variance / min_variance.
COMPUTE balance_ratio = min_group_n / max_group_n.

FORMATS g3_mean g3_median g3_sd g3_variance
 max_variance min_variance hartley_fmax balance_ratio (F10.4).

EXECUTE.

LIST VARIABLES=studytime group_n g3_mean g3_sd g3_variance
 /CASES=FROM 1 TO 4.

LIST VARIABLES=max_variance min_variance hartley_fmax balance_ratio
 /CASES=FROM 1 TO 1.

Hartley F Max Test in Excel

Excel does not need a special Hartley function. You can calculate the statistic directly if each group is placed in a separate column.

Hartley F-Max statistic = MAX(variance_range) / MIN(variance_range)

For this dataset, the Excel calculation should return approximately 1.6800 if the same cleaned data and sample-variance formula are used.

How to Report the Result

A good Hartley F Max Test report should mention the outcome variable, grouping variable, group variances, largest variance, smallest variance, F-Max statistic, assumption warning and final conclusion. Avoid reporting only “Fmax = 1.68” without explaining how it was calculated.

Related Data Analysis Guides

After learning the Hartley F Max Test for variance comparison, continue with related statistical testing guides on Online Internet Cafe: Cochran C Test for detecting the largest variance relative to total variance, Brown Forsythe Test for robust homogeneity-of-variance analysis, Goldfeld Quandt Test for heteroscedasticity in regression, Greenhouse Geisser Correction for repeated-measures ANOVA correction, Durbin Watson Test for residual autocorrelation, DAgostino Pearson Test for normality testing, and Cramer von Mises Test for goodness-of-fit analysis.

When Should You Use This Test?

Use the Hartley F Max Test when you want a simple variance ratio across three or more groups. It is most suitable when the groups are normally distributed and equal or nearly equal in size. It is especially useful as a teaching method because the formula is very easy to understand.

Common Mistakes

1. Using Hartley F Max Test with very unequal group sizes

The classical Hartley test works best when groups are equal or nearly equal. In this example, group sizes are 212, 305, 97 and 35, so the result must be interpreted cautiously.

2. Ignoring normality and outliers

Hartley F-Max is sensitive to non-normal data and outliers. A few extreme values can increase a group variance and make the F-Max statistic look larger than it should.

3. Confusing Hartley F-Max with ordinary ANOVA F

Hartley F-Max is not the same as the ANOVA F statistic. ANOVA F compares between-group mean variation with within-group variation. Hartley F-Max compares the largest group variance with the smallest group variance.

4. Reporting the statistic without the actual variances

A report should show the largest variance, smallest variance, group sample sizes and F-Max value. Without those details, the reader cannot understand the calculation.

5. Treating a non-significant result as proof of equal variances

A non-significant or modest result means there is no strong evidence of unequal variance. It does not prove that all population variances are exactly equal.

Download SPSS Output and Verification Files

The SPSS PDF verifies the main data import, descriptive statistics, group variance calculation, Levene-style homogeneity checks and Hartley F-Max summary.