Quick Answer: Bartlett’s Test Result

The verified SPSS output compares the variance of G3 final grade across the two sex groups, F and M. The female group has N = 383, mean = 12.25, standard deviation = 3.124, and variance = 9.760. The male group has N = 266, mean = 11.41, standard deviation = 3.321, and variance = 11.027. The total sample has N = 649, mean = 11.91, standard deviation = 3.231, and variance = 10.437.

The Bartlett’s Test result reports k = 2 groups, total N = 649, pooled variance = 10.27911, Bartlett chi-square = 1.17082, df = 1, and p = .27923. Since the p-value is greater than .05, we fail to reject the null hypothesis. This means the evidence is not strong enough to conclude that the group variances are significantly different.

Hypothesis-style interpretation: The null hypothesis says the group variances are equal. The alternative hypothesis says at least one group variance is different. Since p = .27923, the test does not find a statistically significant variance difference between the female and male G3 final grade groups. For this output, Bartlett’s Test supports the equal variance assumption.

What Is Bartlett’s Test?

Bartlett’s Test is a homogeneity-of-variance test. It checks whether the variances of a numeric variable are equal across two or more independent groups. In this guide, the numeric variable is G3 final grade, and the grouping variable is sex, with the groups F and M.

The test is commonly used when analysts need to decide whether the equal variance assumption is reasonable before running an analysis that relies on similar group variances. For example, a one-way ANOVA assumes that the dependent variable has roughly equal variances across groups. Bartlett’s Test gives a formal statistical decision about this assumption.

The key idea is simple: if group variances are similar, the test statistic is small and the p-value is usually not significant. If group variances are very different, the test statistic becomes larger and the p-value becomes smaller. In this example, the female variance is 9.760 and the male variance is 11.027. These values are close enough that Bartlett’s Test gives p = .27923, so the equal variance assumption is not rejected.

Bartlett’s Test is strongly connected with other variance and assumption checks. If the data are clearly non-normal, compare its conclusion with Levene’s test, Brown-Forsythe test, Cochran’s C test, and Goldfeld-Quandt test. If you are checking distribution shape before choosing Bartlett’s Test, review the Q-Q plot normality check, P-P plot normality check, Kolmogorov-Smirnov test, Lilliefors test, D’Agostino-Pearson test, and Cramer-von Mises test.

Why Bartlett’s Test Matters

Bartlett’s Test matters because many group-comparison methods are affected by unequal variances. When one group has much larger spread than another, standard tests can produce misleading p-values, confidence intervals and conclusions. Checking variance equality helps you decide whether the ordinary equal-variance method is acceptable or whether a robust alternative is needed.

In the verified SPSS output, the female group has a standard deviation of 3.124, while the male group has a standard deviation of 3.321. The male group is slightly more variable, but the difference is not statistically significant under Bartlett’s Test. Therefore, for this specific comparison of G3 by sex, there is no strong evidence of unequal variances.

For a complete reporting workflow, Bartlett’s Test should be supported by descriptive summaries, boxplots and standard deviation comparisons. Use descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, five-number summary, and coefficient of variation to explain the group spread before reporting the formal test.

Bartlett’s Test Formula

Bartlett’s Test compares the natural logarithm of the pooled variance with the weighted natural logarithms of the group variances. The test statistic is approximately chi-square distributed when the normality assumption is reasonable.

Let k be the number of groups, n_i be the sample size of group i, s_i² be the variance of group i, and N be the total sample size.

1. Pooled Variance

In the verified SPSS output, the pooled variance is:

2. Bartlett’s Test Statistic

The correction factor is:

3. Degrees of Freedom

For this example, k = 2, so df = 1. The verified result is:

Null and Alternative Hypothesis for Bartlett’s Test

Bartlett’s Test is a formal hypothesis test for equality of variances across groups. In this post, it compares the variance of G3 final grade across the female and male groups.

Dataset and Variables Used

The worked example uses a student performance-style dataset. The dependent or measured variable is G3 final grade, and the grouping variable is sex. Bartlett’s Test checks whether the spread of G3 is statistically similar for female and male students.

Before applying Bartlett’s Test, always inspect the group distributions. In this example, the normality context is important because the SPSS normality tests are significant in both groups. Use normality and shape guides such as Q-Q plot normality check, P-P plot normality check, histogram interpretation, and box plot interpretation before relying only on the Bartlett p-value.

Verified SPSS Output Interpretation

The verified SPSS output provides the group descriptives, normality context, Bartlett variance components and final equal-variance decision. The output compares G3 final grade by sex. There are 649 included cases, 0 excluded cases, and 649 total cases.

The SPSS output PDF for this guide is available here:

SPSS Group Descriptives

SPSS Normality Context

SPSS Bartlett’s Test Result

SPSS Decision Summary

How to Handle the Normality Warning

The normality context shows significant normality tests in both groups. Because Bartlett’s Test is sensitive to non-normality, the safest reporting strategy is to state both findings: Bartlett’s Test did not find unequal variances, but because group distributions departed from normality, a robust variance test such as Levene’s test or Brown-Forsythe test should also be considered. This balanced interpretation is stronger than reporting Bartlett’s p-value alone.

Python Chart-by-Chart Interpretation

The Python charts show Bartlett’s Test visually. They explain group spread, group variance comparison, p-value decision, normality context and group size with standard deviation. These visuals help turn the variance test into a clear report.

Python Chart 1: Group Spread Boxplots

This chart shows the distribution of G3 final grades by group. Boxplots are useful before Bartlett’s Test because they show the median, interquartile range, overall spread and possible unusual values. If one group’s box or whiskers were much wider than the other group’s, that would visually suggest unequal variance.

In this example, the female and male spreads are broadly similar. The male group has a slightly larger standard deviation and variance, but the difference is not visually extreme. This matches the SPSS result where the female variance is 9.760 and the male variance is 11.027.

Decision/reporting conclusion: The boxplots support the Bartlett’s Test decision. The group spreads do not appear dramatically different, and the formal test does not reject equal variances.

Python Chart 2: Group Variance Comparison

This chart directly compares the variance values used in Bartlett’s Test. The female group variance is 9.760, and the male group variance is 11.027. The male variance is numerically higher, but the difference is modest.

Bartlett’s Test evaluates whether this observed variance difference is large enough to be statistically significant. The result, χ²(1) = 1.17082, p = .27923, shows that the difference is not statistically significant at the .05 level.

Decision/reporting conclusion: Report that the variance comparison chart shows only a modest variance difference, and Bartlett’s Test confirms that the difference is not statistically significant.

Python Chart 3: Bartlett p-value Decision

This chart presents the most important decision visually. Bartlett’s Test gives a p-value of .27923. Since this value is greater than the common alpha level of .05, the correct decision is to fail to reject the null hypothesis of equal variances.

A p-value chart is useful for student reports because it makes the decision rule obvious. The p-value sits above the rejection boundary, so the test does not provide evidence that the group variances are different.

Decision/reporting conclusion: Report that Bartlett’s Test was not statistically significant, p = .27923. The equal variance assumption is acceptable according to Bartlett’s Test.

Python Chart 4: Normality Context Distribution

This chart is important because Bartlett’s Test assumes that the group distributions are approximately normal. The SPSS normality context shows that both groups reject normality: female group Shapiro-Wilk W = .934, p < .001, and male group Shapiro-Wilk W = .913, p < .001.

Because of this, the Bartlett result should be interpreted carefully. The non-significant Bartlett result still suggests no strong variance difference, but robust tests are recommended as a confirmation. This is where Levene’s test and Brown-Forsythe test become useful.

Decision/reporting conclusion: Report that the normality context warns against relying only on Bartlett’s Test. The result is non-significant, but robust variance tests should be considered because group normality is not supported.

Python Chart 5: Group Size and Standard Deviation

This chart combines group size and spread. The female group has N = 383 and SD = 3.124, while the male group has N = 266 and SD = 3.321. The group sizes are not identical, but both groups are large enough for stable descriptive summaries.

The standard deviations are close, which supports the non-significant Bartlett result. If the standard deviations were very different, the p-value decision might be different, especially with large sample sizes.

Decision/reporting conclusion: Report that the group standard deviations are similar and support the equal variance conclusion. The male group is slightly more variable, but not significantly so under Bartlett’s Test.

R Chart-by-Chart Validation

The R charts validate the same Bartlett’s Test workflow using a separate statistical environment. Agreement between SPSS, Python and R strengthens the conclusion that the group variances are not significantly different.

R Chart 1: Group Spread Boxplots

The R boxplot chart validates the Python boxplot interpretation. It shows that the female and male distributions have broadly comparable spread. Any difference in spread appears modest rather than extreme.

Decision/reporting conclusion: Report that the R boxplots confirm the Python visual conclusion: the two groups do not show a large visual variance difference.

R Chart 2: Group Variance Comparison

The R variance comparison chart confirms that the male group variance is slightly larger than the female group variance, but the difference is not large enough to reject equal variances. This visual result aligns with χ²(1) = 1.17082, p = .27923.

Decision/reporting conclusion: Report that R validates the same modest variance difference seen in Python and SPSS. The equal variance assumption is not rejected.

R Chart 3: Bartlett p-value Decision

The R p-value decision chart confirms the formal Bartlett conclusion. Because p = .27923 is greater than .05, the test result is not statistically significant. The null hypothesis of equal variances is not rejected.

Decision/reporting conclusion: Report that the R decision chart validates the Python and SPSS decision. Bartlett’s Test supports equal variances for G3 across sex groups.

R Chart 4: Normality Context Distribution

The R normality context chart confirms that Bartlett’s Test should be interpreted with caution when group distributions deviate from normality. The SPSS normality tests are significant in both groups, so the result should be complemented with more robust variance checks.

Decision/reporting conclusion: Report that the R normality context supports the same caution as Python and SPSS. Bartlett’s Test is non-significant, but Levene or Brown-Forsythe can be reported as robustness checks when distributions are non-normal.

R Chart 5: Group Size and Standard Deviation

The R group size and standard deviation chart validates the descriptive basis of the test. The female group is larger, but both groups have substantial sample sizes. The standard deviations are close: 3.124 for female students and 3.321 for male students.

Decision/reporting conclusion: Report that R confirms the descriptive similarity in group spread. The standard deviation difference is small and consistent with the non-significant Bartlett result.

SPSS, R, Python and Excel Workflows for Bartlett’s Test

The Bartlett’s Test workflow is simple across statistical software. First, identify the numeric dependent variable and the independent grouping variable. Second, inspect group descriptive statistics and distribution shape. Third, run Bartlett’s Test. Fourth, interpret the chi-square statistic, degrees of freedom and p-value. Finally, report whether the equal variance assumption is acceptable.

SPSS Workflow

Python Workflow

R Workflow

Excel Workflow

Code Blocks for Bartlett’s Test

SPSS Syntax for Bartlett’s Test Support

* Bartlett's Test support in SPSS.
* Example: G3 final grade by sex group.

TITLE "Bartlett's Test: Homogeneity of Variance for G3 by Sex".

DATASET ACTIVATE DataSet1.

* Group descriptives.
MEANS TABLES=G3 BY sex
  /CELLS=COUNT MEAN STDDEV VARIANCE.

* Normality context by group.
EXAMINE VARIABLES=G3 BY sex
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

* If Bartlett components are already calculated in a grouped table,
* report the verified result:
* k = 2, total N = 649, pooled variance = 10.27911,
* Bartlett chi-square = 1.17082, df = 1, p = .27923.

* Robust context: Levene-style test is available through ONEWAY
* when the grouping variable is numeric or properly recoded.
* Example after recoding sex to numeric group:
* ONEWAY G3 BY sex_num
*   /STATISTICS HOMOGENEITY
*   /MISSING ANALYSIS.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Bartletts-Test-SPSS-Output.pdf".

Python Code for Bartlett’s Test

import pandas as pd
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

# Load data
df = pd.read_csv("dataset.csv")

# Prepare variables
df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df = df.dropna(subset=["G3", "sex"])

# Split groups
groups = [g["G3"].values for name, g in df.groupby("sex")]
group_names = list(df.groupby("sex").groups.keys())

# Group summaries
summary = df.groupby("sex")["G3"].agg(["count", "mean", "std", "var"])
print(summary)

# Bartlett's Test
bart_stat, bart_p = stats.bartlett(*groups)
print("Bartlett chi-square:", bart_stat)
print("Bartlett p-value:", bart_p)

# Decision
alpha = 0.05
if bart_p < alpha:
    print("Reject H0: group variances are significantly different.")
else:
    print("Fail to reject H0: no significant evidence of unequal variances.")

# Chart 1: Group spread boxplots
plt.figure(figsize=(8, 6))
plt.boxplot(groups, labels=group_names)
plt.xlabel("Group")
plt.ylabel("G3 final grade")
plt.title("Group Spread Boxplots for Bartlett's Test")
plt.tight_layout()
plt.savefig("bartlett-s-test-python-01-chart_01_group_spread_boxplots.png", dpi=300)
plt.close()

# Chart 2: Group variance comparison
plt.figure(figsize=(8, 6))
plt.bar(summary.index.astype(str), summary["var"])
plt.xlabel("Group")
plt.ylabel("Variance")
plt.title("Group Variance Comparison")
plt.tight_layout()
plt.savefig("bartlett-s-test-python-02-chart_02_group_variance_comparison.png", dpi=300)
plt.close()

# Chart 3: Bartlett p-value decision
plt.figure(figsize=(8, 5))
plt.bar(["Bartlett p-value"], [bart_p])
plt.axhline(alpha, linestyle="--")
plt.ylabel("p-value")
plt.title("Bartlett p-value Decision")
plt.tight_layout()
plt.savefig("bartlett-s-test-python-03-chart_03_bartlett_p_value_decision.png", dpi=300)
plt.close()

# Chart 4: Normality context distribution
plt.figure(figsize=(8, 6))
for name, g in df.groupby("sex"):
    plt.hist(g["G3"], bins=15, alpha=0.5, label=str(name), density=True)
plt.xlabel("G3 final grade")
plt.ylabel("Density")
plt.title("Normality Context Distribution")
plt.legend()
plt.tight_layout()
plt.savefig("bartlett-s-test-python-04-chart_04_normality_context_distribution.png", dpi=300)
plt.close()

# Chart 5: Group size and standard deviation
fig, ax1 = plt.subplots(figsize=(8, 6))
ax1.bar(summary.index.astype(str), summary["count"], alpha=0.7)
ax1.set_xlabel("Group")
ax1.set_ylabel("Sample size")
ax2 = ax1.twinx()
ax2.plot(summary.index.astype(str), summary["std"], marker="o")
ax2.set_ylabel("Standard deviation")
plt.title("Group Size and Standard Deviation")
fig.tight_layout()
plt.savefig("bartlett-s-test-python-05-chart_05_group_size_and_standard_deviation.png", dpi=300)
plt.close()

R Code for Bartlett’s Test

# Bartlett's Test in R

library(ggplot2)
library(dplyr)

df <- read.csv("dataset.csv")

df$G3 <- as.numeric(df$G3)
df$sex <- as.factor(df$sex)

df_clean <- df %>%
  filter(!is.na(G3), !is.na(sex))

# Group summaries
summary_table <- df_clean %>%
  group_by(sex) %>%
  summarise(
    n = n(),
    mean = mean(G3),
    sd = sd(G3),
    variance = var(G3),
    .groups = "drop"
  )

print(summary_table)

# Bartlett's Test
bart_result <- bartlett.test(G3 ~ sex, data = df_clean)
print(bart_result)

# Decision
alpha <- 0.05
if (bart_result$p.value < alpha) {
  cat("Reject H0: group variances are significantly different.\n")
} else {
  cat("Fail to reject H0: no significant evidence of unequal variances.\n")
}

# Chart 1: Group spread boxplots
ggplot(df_clean, aes(x = sex, y = G3)) +
  geom_boxplot() +
  labs(title = "Group Spread Boxplots for Bartlett's Test",
       x = "Group", y = "G3 final grade")

# Chart 2: Group variance comparison
ggplot(summary_table, aes(x = sex, y = variance)) +
  geom_col() +
  labs(title = "Group Variance Comparison",
       x = "Group", y = "Variance")

# Chart 3: Bartlett p-value decision
p_df <- data.frame(test = "Bartlett p-value", p_value = bart_result$p.value)

ggplot(p_df, aes(x = test, y = p_value)) +
  geom_col() +
  geom_hline(yintercept = alpha, linetype = "dashed") +
  labs(title = "Bartlett p-value Decision",
       x = "", y = "p-value")

# Chart 4: Normality context distribution
ggplot(df_clean, aes(x = G3, fill = sex)) +
  geom_histogram(aes(y = after_stat(density)), bins = 15, alpha = 0.5, position = "identity") +
  labs(title = "Normality Context Distribution",
       x = "G3 final grade", y = "Density")

# Chart 5: Group size and standard deviation
ggplot(summary_table, aes(x = sex)) +
  geom_col(aes(y = n)) +
  geom_point(aes(y = sd * max(n) / max(sd)), size = 3) +
  geom_line(aes(y = sd * max(n) / max(sd), group = 1)) +
  labs(title = "Group Size and Standard Deviation",
       x = "Group", y = "Sample size with scaled SD overlay")

Excel Formulas for Bartlett’s Test

Assume:
Female G3 values are in A2:A384.
Male G3 values are in B2:B267.

Female sample size:
=COUNT(A2:A384)

Male sample size:
=COUNT(B2:B267)

Female variance:
=VAR.S(A2:A384)

Male variance:
=VAR.S(B2:B267)

Total N:
=COUNT(A2:A384)+COUNT(B2:B267)

Number of groups:
=2

Pooled variance:
=((n1-1)*var1+(n2-1)*var2)/(N-k)

Correction factor:
=1+(1/(3*(k-1)))*((1/(n1-1)+1/(n2-1))-(1/(N-k)))

Bartlett chi-square:
=(((N-k)*LN(pooled_variance))-((n1-1)*LN(var1)+(n2-1)*LN(var2)))/correction_factor

Degrees of freedom:
=k-1

p-value:
=CHISQ.DIST.RT(bartlett_chisq,df)

Decision:
If p < .05, reject equal variances.
If p >= .05, fail to reject equal variances.

APA Reporting Wording for Bartlett’s Test

When reporting Bartlett’s Test, include the tested variable, grouping variable, group variances, test statistic, degrees of freedom, p-value and final equal-variance decision. Also mention the normality context if the group distributions are non-normal.

APA-Style Report

Short Report Sentence

Normality-Cautious Report Sentence

Student-Friendly Report Example

Common Mistakes in Bartlett’s Test Interpretation

When to Use Bartlett’s Test

Use Bartlett’s Test when you need a formal test of equal variances across two or more independent groups and the group distributions are approximately normal. It is especially useful as an assumption check before ANOVA or other equal-variance procedures.

If Bartlett’s Test is significant, the equal variance assumption may be violated. Depending on the analysis, you may need Welch’s ANOVA, robust standard errors, transformations or nonparametric alternatives. For related assumption and reporting decisions, see confidence interval, effect size, one-tailed t-test, one-sample z-test, one-proportion z-test, and cross tabulation.

Downloads and Resources for Bartlett’s Test

The resources below include the verified SPSS output PDF, Python charts and R validation charts used in this guide. These files support direct reporting, teaching, WordPress publishing and variance-assumption diagnostics practice.

Related Statistical Guides

Use these related Salar Cafe guides to connect Bartlett’s Test with equal-variance checks, normality testing, descriptive statistics, transformations, hypothesis testing, confidence intervals and reporting.