Quick Answer: Bonferroni Correction Result

The Bonferroni Correction was applied after a significant one-way ANOVA comparing G3 final grade across four studytime groups. Because there are four groups, there are 6 pairwise comparisons. With a familywise alpha of .05, the Bonferroni per-comparison alpha is .008333.

The one-way ANOVA result was significant, F(3, 645) = 15.876, p = 5.705728e-10. The Bonferroni-adjusted pairwise results show that studytime 1 differs from studytime 2, studytime 1 differs from studytime 3, studytime 1 differs from studytime 4, and studytime 2 differs from studytime 3. The comparisons studytime 2 vs 4 and studytime 3 vs 4 are not significant after Bonferroni correction.

What Is Bonferroni Correction?

Bonferroni Correction is a multiple-comparison adjustment used when a researcher tests several hypotheses in the same family of comparisons. In post hoc ANOVA analysis, this usually means comparing every pair of group means after the overall ANOVA has shown that at least one group mean differs.

The problem is simple. If you run one test at α = .05, the chance of a false positive is controlled at about 5% for that test. But if you run many tests, the chance of getting at least one false positive increases. Bonferroni correction solves this by making each individual comparison harder to call significant.

In this example, there are four studytime groups, so there are six pairwise comparisons. Bonferroni correction either divides the familywise alpha by the number of comparisons or multiplies each raw p-value by the number of comparisons. Both views lead to the same decision logic.

This guide connects naturally with One Way ANOVA, Factorial ANOVA, ANOVA Assumptions, ANOVA in SPSS, ANOVA in R, ANOVA in Python, P Value, Null and Alternative Hypothesis, Type I and Type II Error and Statistical Power.

Bonferroni Correction Formula

The Bonferroni correction can be written in two common ways. The first method divides the familywise alpha by the number of comparisons:

Here, m is the number of pairwise comparisons. In this example, there are four groups, so the number of comparisons is:

With familywise α = .05:

The second method adjusts each p-value directly:

SPSS, Python and R commonly report Bonferroni-adjusted p-values. A pairwise comparison is significant when the adjusted p-value is below .05.

Why Bonferroni Correction Is Needed After ANOVA

A one-way ANOVA tells whether at least one group mean differs from the others. It does not tell exactly which groups differ. After a significant ANOVA, a post hoc test is used to compare group pairs. When many pairwise comparisons are made, the false-positive risk increases, so a correction is needed.

In this example, the ANOVA result is significant, but the meaningful reporting question is more specific: which studytime groups have different mean G3 scores? Bonferroni correction answers that question while controlling the familywise error rate.

For broader mean-comparison context, review T Test vs ANOVA, Student’s T Test, Two Sample T Test, One Tailed T Test, Two Tailed T Test, T Test in SPSS, T Test in R and T Test in Python.

Bonferroni Post Hoc Hypotheses

Bonferroni post hoc analysis tests pairwise hypotheses. Each pair has its own null and alternative hypothesis, but the p-values are adjusted so the family of tests is controlled.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The post hoc factor is studytime, which has four groups. The analysis uses 649 valid cases.

Group Descriptive Statistics

The descriptive pattern explains why post hoc testing is useful. The mean rises from studytime 1 to studytime 3, while studytime 4 remains high but has fewer students. Bonferroni correction then identifies which of these visible mean differences remain statistically significant after controlling for multiple testing.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Five Number Summary and Box Plot Interpretation.

Assumptions Before Bonferroni Post Hoc Testing

Bonferroni Correction adjusts for multiple comparisons, but it does not fix a poor ANOVA design. The same basic assumptions of one-way ANOVA should be considered before interpreting Bonferroni post hoc results.

For assumption support, use ANOVA Assumptions, Levene Test, Bartlett’s Test, Brown-Forsythe Test, Hartley F Max Test, Cochran C Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test and Outlier Detection.

SPSS Output Interpretation for Bonferroni Correction

The SPSS output uses ONEWAY G3 BY studytime with descriptive statistics, homogeneity testing, ANOVA and Bonferroni post hoc comparisons. The key table is the Multiple Comparisons table under Bonferroni.

SPSS Reading Order

SPSS One-Way ANOVA Table

SPSS Bonferroni Multiple Comparisons

Python Chart-by-Chart Interpretation

The Python chart sequence explains Bonferroni Correction through group distributions, group means, adjusted p-values, mean differences with Bonferroni confidence intervals and group sample sizes.

Python Chart 1: Group Distributions by Studytime

The first chart shows how G3 final grade is distributed within each studytime group. Studytime group 1 has the lowest central pattern, while studytime groups 3 and 4 sit higher on the grade scale. The visible group separation explains why the one-way ANOVA becomes significant.

The boxplot also shows that the groups are not identical in spread or sample behavior. Bonferroni correction does not remove the need to inspect distributions; it controls the multiple-testing error after the ANOVA and pairwise comparisons are run.

Python Chart 2: Group Means with Confidence Intervals

The group mean chart shows a clear increase from studytime 1 to studytime 3. Group 4 remains high, but it has the smallest sample size, which makes its interval wider than the larger groups.

This visual supports the post hoc result. The largest corrected differences involve group 1 compared with higher studytime groups, especially group 3.

Python Chart 3: Bonferroni-Adjusted p-values

The adjusted p-value chart places each comparison against the α = .05 decision line. The comparisons 1 vs 3, 1 vs 2, 1 vs 4 and 2 vs 3 fall on the significant side after Bonferroni correction.

The comparisons 2 vs 4 and 3 vs 4 remain to the right of the alpha line and are not significant after correction. This means the highest studytime groups are not all statistically separated from each other after familywise error control.

Python Chart 4: Bonferroni Mean Differences with Confidence Intervals

The mean-difference chart shows which corrected intervals cross zero. Comparisons whose intervals do not cross zero are significant after Bonferroni correction. The differences involving group 1 and the comparison 2 vs 3 do not cross zero.

The intervals for 2 vs 4 and 3 vs 4 cross zero. This matches the adjusted p-value result and confirms that those two comparisons should be reported as not significant after correction.

Python Chart 5: Group Sample Sizes

The sample-size chart shows that the groups are unequal: group 2 has 305 cases, group 1 has 212, group 3 has 97 and group 4 has 35. This matters because smaller groups usually have larger standard errors in pairwise comparisons.

Group 4 has a high mean but only 35 cases. That smaller sample size helps explain why some comparisons involving group 4 are less precise and why the 2 vs 4 comparison does not remain significant after correction.

R Chart-by-Chart Validation

The R charts validate the same Bonferroni Correction workflow using a second software environment. The R results confirm the same group distribution, mean pattern, adjusted p-value decision, confidence interval interpretation and sample-size context.

R Chart 1: Group Distributions by Studytime

The R distribution chart confirms the same pattern shown by Python. Studytime group 1 has a lower grade distribution, while groups 3 and 4 appear higher.

This validates the descriptive foundation for the post hoc test. The statistical comparisons are not floating numbers; they are attached to a visible group pattern.

R Chart 2: Group Means with Confidence Intervals

The R mean chart confirms that studytime group 1 has the lowest mean and group 3 has the highest mean. Group 4 remains close to group 3 but with a wider interval because its sample size is smaller.

This supports the final interpretation that the strongest differences involve lower studytime compared with higher studytime groups.

R Chart 3: Bonferroni-Adjusted p-values

The R adjusted p-value chart confirms the Python decision pattern. The comparisons 1 vs 2, 1 vs 3, 1 vs 4 and 2 vs 3 are significant after correction.

The comparisons 2 vs 4 and 3 vs 4 are not significant after correction. This repeated result across R, Python and SPSS makes the post hoc conclusion stable for reporting.

R Chart 4: Bonferroni Mean Differences with Confidence Intervals

The R mean-difference chart confirms that the significant comparisons have intervals away from zero. The non-significant comparisons have intervals that cross zero.

This chart is useful because it gives both direction and uncertainty. A negative difference such as 1 vs 3 means group 1 has a lower mean than group 3.

R Chart 5: Group Sample Sizes

The R sample-size chart confirms the unequal group sizes. Group 2 is the largest group, and group 4 is the smallest group.

This context helps explain why group 4 comparisons need careful interpretation. Bonferroni correction controls the familywise error rate, while the sample-size chart explains the precision behind the pairwise estimates.

SPSS, R, Python and Excel Workflows for Bonferroni Correction

The same Bonferroni Correction workflow can be reproduced in SPSS, R, Python and Excel. SPSS provides Bonferroni directly in the One-Way ANOVA post hoc menu. R can use pairwise.t.test() with p.adjust.method = "bonferroni". Python can calculate adjusted p-values by multiplying raw p-values by the number of comparisons. Excel can support the calculation with formulas, but SPSS, R or Python is better for final reproducible reporting.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Bonferroni Correction

SPSS Syntax for Bonferroni Correction

* Bonferroni Correction post hoc test in SPSS.
* Dependent variable: G3.
* Factor: studytime.

TITLE "Bonferroni Correction Post Hoc Analysis".

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY
  /POSTHOC = BONFERRONI ALPHA(.05)
  /MISSING ANALYSIS.

EXAMINE VARIABLES=G3 BY studytime
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Bonferroni-Correction-SPSS-Output.pdf".

Python Code for Bonferroni Correction

import pandas as pd
import numpy as np
from itertools import combinations
from scipy import stats

df = pd.read_csv("dataset.csv")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")

data = df[["G3", "studytime"]].dropna().copy()

# Group summary
summary = data.groupby("studytime")["G3"].agg(
    n="count",
    mean="mean",
    standard_deviation="std",
    variance="var",
    standard_error=lambda x: x.std(ddof=1) / np.sqrt(x.count()),
    minimum="min",
    maximum="max"
).reset_index()

print(summary)

# One-way ANOVA
groups = [
    group["G3"].to_numpy()
    for _, group in data.groupby("studytime", observed=True)
]

anova = stats.f_oneway(*groups)
print("ANOVA F:", anova.statistic)
print("ANOVA p:", anova.pvalue)

# Bonferroni pairwise comparisons
labels = sorted(data["studytime"].dropna().unique())
pairs = list(combinations(labels, 2))
m = len(pairs)

results = []

for g1, g2 in pairs:
    x1 = data.loc[data["studytime"] == g1, "G3"].to_numpy()
    x2 = data.loc[data["studytime"] == g2, "G3"].to_numpy()

    t_stat, raw_p = stats.ttest_ind(x1, x2, equal_var=True)
    adj_p = min(raw_p * m, 1.0)

    mean_diff = x1.mean() - x2.mean()

    results.append({
        "group_1": g1,
        "group_2": g2,
        "mean_group_1": x1.mean(),
        "mean_group_2": x2.mean(),
        "mean_difference_group_1_minus_group_2": mean_diff,
        "t_value": t_stat,
        "raw_p_value": raw_p,
        "bonferroni_adjusted_p_value": adj_p,
        "decision_alpha_0_05": "Significant after Bonferroni correction"
        if adj_p < 0.05 else "Not significant after Bonferroni correction"
    })

pairwise_table = pd.DataFrame(results)
print(pairwise_table)

R Code for Bonferroni Correction

# Bonferroni Correction post hoc analysis in R

library(tidyverse)
library(car)

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

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

data <- df %>%
  select(G3, studytime) %>%
  drop_na()

# Group summary
data %>%
  group_by(studytime) %>%
  summarise(
    n = n(),
    mean_G3 = mean(G3),
    sd_G3 = sd(G3),
    variance_G3 = var(G3),
    se_G3 = sd(G3) / sqrt(n()),
    minimum = min(G3),
    maximum = max(G3),
    .groups = "drop"
  )

# Assumption context
leveneTest(G3 ~ studytime, data = data)

# One-way ANOVA
anova_model <- aov(G3 ~ studytime, data = data)
summary(anova_model)

# Bonferroni pairwise comparisons
pairwise.t.test(
  x = data$G3,
  g = data$studytime,
  p.adjust.method = "bonferroni",
  pool.sd = TRUE
)

# Optional: Tukey-style table with Bonferroni-adjusted p-values
pairwise_results <- pairwise.t.test(
  x = data$G3,
  g = data$studytime,
  p.adjust.method = "bonferroni",
  pool.sd = TRUE
)

print(pairwise_results)

Excel Notes for Bonferroni Correction

Excel support workflow:

1. Arrange the data:
   G3 | studytime

2. Count number of groups:
   k = 4

3. Count pairwise comparisons:
   =k*(k-1)/2
   =4*(4-1)/2
   =6

4. Calculate Bonferroni corrected alpha:
   =0.05/6
   =0.008333

5. Run pairwise t-tests for all six comparisons:
   1 vs 2
   1 vs 3
   1 vs 4
   2 vs 3
   2 vs 4
   3 vs 4

6. Adjust each p-value:
   =MIN(raw_p_value*6,1)

7. Decision:
   =IF(adjusted_p_value<0.05,"Significant","Not significant")

8. Report:
   Mean difference, raw p-value, adjusted p-value, corrected decision.

APA Reporting Wording

When reporting Bonferroni Correction, first report the omnibus ANOVA result, then report the corrected pairwise comparisons. Mention that Bonferroni adjustment was used to control familywise error.

Common Mistakes

When to Use Bonferroni Correction

Use Bonferroni Correction when you need a simple and conservative way to control familywise error across multiple tests. It is often used after one-way ANOVA, factorial ANOVA, repeated-measures comparisons and planned pairwise comparisons.

Compare this guide with One Way ANOVA, Factorial ANOVA, Mixed ANOVA, Nested ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA Effect Size, F Distribution, Eta Squared and Cohen’s F Formula.

Downloads and Resources for Bonferroni Correction

Use these resources to reproduce the Bonferroni Correction workflow. The Python report, R report and SPSS output PDF are included as verification files. Script and workbook placeholders can be replaced after the final downloadable files are uploaded to the WordPress Media Library.

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