Quick Answer: Welch’s ANOVA Result

The Welch’s ANOVA result compares mean G3 final grade across four studytime groups without relying on the strict equal-variance assumption used by ordinary one-way ANOVA. The group means are approximately 10.84, 12.09, 13.23 and 13.06. The result is statistically significant: Welch F(3, 139.10) = 18.183, p = 5.188e-10.

The decision is to reject the null hypothesis of equal group means. Students in different studytime groups do not have the same average G3 final grade in this dataset. The mean pattern increases from studytime 1 to studytime 3, while studytime 4 remains high but has a much smaller sample size.

What Is Welch’s ANOVA?

Welch’s ANOVA is a robust version of one-way ANOVA that compares group means while adjusting for unequal variances and unequal group sizes. Ordinary one-way ANOVA pools the group variances into one common error term. Welch’s ANOVA does not depend on that same equal-variance assumption, so it is safer when variances are unequal or sample sizes differ strongly across groups.

In this example, the dependent variable is G3 final grade, and the grouping factor is studytime with four groups. The studytime group sizes are unequal: 212, 305, 97 and 35. Welch’s ANOVA uses group-specific variance and sample-size information to build a robust F statistic and adjusted denominator degrees of freedom.

The important result is that the Welch test is significant. This means the four studytime groups do not have equal mean G3 scores. It does not automatically tell which groups differ from each other. For group-to-group comparisons after Welch’s ANOVA, a Games-Howell post hoc test is usually more appropriate than Tukey when equal variances are not assumed.

This guide connects naturally with One Way ANOVA, ANOVA Assumptions, Brown Forsythe ANOVA, Brown-Forsythe Test, Levene Test, Bartlett’s Test, F Distribution, ANOVA Effect Size and Eta Squared.

Welch’s ANOVA Formula

The ordinary one-way ANOVA F statistic uses a pooled within-group variance. Welch’s ANOVA instead gives each group a weight based on its sample size and variance. Groups with larger sample sizes and smaller variances receive more weight in the adjusted test.

Here, w_i is the Welch weight for group i, n_i is the group sample size, and s_i² is the group variance. The weighted mean is calculated as:

The Welch F statistic then compares weighted group mean differences with an adjusted error structure. The denominator degrees of freedom are not the same as ordinary one-way ANOVA. In this example, the denominator df is 139.10, not 645.

Welch Weights in This Example

Decision Formula

In this example, p = 5.188e-10, which is far below α = .05. Therefore, the null hypothesis of equal studytime group means is rejected.

Classic One-Way ANOVA vs Welch’s ANOVA

Classic one-way ANOVA and Welch’s ANOVA both test whether group means differ. The difference is how they treat variance and sample size. Classic ANOVA assumes a common within-group variance. Welch’s ANOVA adjusts the test when variances and sample sizes are not perfectly equal.

Welch’s ANOVA Hypotheses

Welch’s ANOVA tests whether all group means are equal. The hypothesis is the same general mean-comparison hypothesis as one-way ANOVA, but the test statistic and denominator degrees of freedom are adjusted.

Dataset and Variables Used

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

Descriptive Statistics

The descriptive table shows the same story as the charts. Mean G3 increases from studytime 1 to studytime 3. Studytime 4 remains high, but the group contains only 35 students, so its confidence interval is wider.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, P Value and Null and Alternative Hypothesis.

Assumptions for Welch’s ANOVA

Welch’s ANOVA relaxes the equal-variance assumption, but it does not remove all assumptions. The dependent variable should be numeric, the groups should be independent, and the observations should be reasonably independent within groups. The method is more robust than ordinary one-way ANOVA when variances and sample sizes are unequal.

The SPSS Levene table does not reject equal variance in this dataset, with p = .400 based on the mean and p = .380 based on the median. This does not make Welch’s ANOVA wrong. It simply means Welch is being used as a robust confirmation rather than because the variance test strongly failed.

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 and Outlier Detection.

SPSS Output Interpretation for Welch’s ANOVA

The SPSS output uses ONEWAY G3 BY studytime with /STATISTICS DESCRIPTIVES HOMOGENEITY WELCH. This produces the ordinary one-way ANOVA table, the test of homogeneity of variances, and the robust Welch test of equality of means.

SPSS Reading Order

SPSS Descriptive Statistics

SPSS Homogeneity and ANOVA Results

Python Chart-by-Chart Interpretation

The Python chart sequence explains Welch’s ANOVA through group means, distribution shape, the observed Welch F statistic, p-value decision, variance context, group weights, group size with standard deviation, and the final summary table.

Python Chart 1: Welch’s Group Means with 95% Confidence Intervals

The first chart shows that mean G3 rises from studytime group 1 to studytime group 3. Studytime group 1 has the lowest average final grade, while studytime group 3 has the highest average final grade. Studytime group 4 remains high but has a wider interval because its sample size is much smaller.

This chart explains why Welch’s ANOVA rejects equal means. The group means are not flat across studytime levels. The visible pattern supports a practical conclusion: students in higher studytime groups tend to have higher G3 final grades.

Python Chart 2: Distribution by Studytime Group

The distribution chart shows central tendency, spread and possible outliers within each studytime group. Studytime group 1 has a lower distribution, while studytime groups 3 and 4 show higher central values.

The chart is important because Welch’s ANOVA is often used when group spreads may differ. Here, group 3 has a smaller standard deviation than groups 1 and 2, while group 4 is small and has a wider confidence interval.

Python Chart 3: Observed Welch F on F Distribution

The observed Welch F statistic is 18.183, plotted far into the right tail of the F distribution. The chart subtitle reports df1 = 3, df2 = 139.10, and p = 5.188e-10.

This visual makes the hypothesis decision clear. The observed F statistic is much larger than what would be expected if all four group means were equal, so the null hypothesis is rejected.

Python Chart 4: Welch’s ANOVA p-value Decision

The p-value chart compares α = .05, Welch p = 5.188e-10, and classic ANOVA p = 5.706e-10. Both p-values are far below .05.

This means the conclusion is stable in this dataset. Whether the analyst reads the ordinary one-way ANOVA table or the Welch robust table, the decision is the same: reject equal studytime group means.

Python Chart 5: Variance Context by Group

The variance chart shows that groups 1 and 2 have the largest sample variances, group 4 has a moderate variance, and group 3 has the smallest variance. This variance context helps explain why Welch weights are useful.

In this dataset, the formal Levene test does not reject equal variances, but the variance chart still explains the logic of Welch’s ANOVA. Welch gives different groups different weights based on sample size and variance rather than treating all group variances as one common pooled value.

Python Chart 6: Welch’s Group Weights

The group weight chart shows that studytime group 2 receives the largest Welch weight because it has the largest sample size. Group 1 receives the second-largest weight. Group 3 receives a moderate weight because its variance is smaller, even though its sample size is lower than groups 1 and 2.

Studytime group 4 receives the smallest weight because it has only 35 cases. This is a central feature of Welch’s ANOVA: smaller and more variable groups contribute less weight to the adjusted test.

Python Chart 7: Group Size and Standard Deviation

The group size and standard deviation chart shows why Welch’s ANOVA is useful for real datasets. Group sizes are not equal: group 2 is the largest, group 1 is also large, group 3 is smaller, and group 4 is much smaller.

The standard deviation line shows that group spread is not identical across groups. Welch’s ANOVA adjusts the test to account for this combination of unequal group size and group variability.

Python Chart 8: Welch’s ANOVA Summary Table

The summary table gives the final Python result: Welch F(3, 139.10) = 18.183, p = 5.188e-10. The decision is to reject equal group means.

The same table also shows the group means, standard deviations, variances and weights. This is the best single figure for reporting because it connects the inferential decision with the group-level descriptive statistics.

R Chart-by-Chart Validation

The R charts validate the Python and SPSS results using a second workflow. The R sequence confirms the same mean pattern, distribution pattern, Welch F decision, p-value decision, variance context, group weights, and final summary interpretation.

R Chart 1: Welch’s Group Means with Confidence Intervals

The R group mean chart confirms the same upward pattern across studytime groups. The lowest mean appears in studytime group 1, while the highest mean appears around studytime group 3.

This validates the Python chart and supports the conclusion that the significant Welch result reflects real differences in group means.

R Chart 2: Distribution by Group

The R boxplots confirm the same distribution story. Higher studytime groups show higher central G3 values, while lower studytime groups include lower scores.

This chart supports using Welch’s ANOVA as part of an assumption-aware workflow rather than relying only on a single p-value.

R Chart 3: Observed Welch F Distribution

The R F distribution chart confirms that the observed Welch statistic is far into the right tail. The visual decision is the same as in Python: the statistic is too large to support equal group means.

This software-to-software agreement strengthens confidence in the statistical conclusion.

R Chart 4: p-value Decision

The R p-value chart confirms that the Welch p-value is far below .05. The null hypothesis of equal studytime group means is rejected.

This chart is useful for readers because it shows the decision visually without requiring them to read a dense ANOVA table first.

R Chart 5: Variance Context by Group

The R variance chart confirms that variance is not identical across groups. Group 3 has the smallest variance, while groups 1 and 2 have larger variances.

This validates the Welch weighting explanation and helps readers understand why Welch’s ANOVA is a robust alternative to ordinary one-way ANOVA.

R Chart 6: Welch’s Group Weights

The R group weight chart confirms that studytime group 2 receives the largest weight, while studytime group 4 receives the smallest weight.

This reinforces the main technical point of Welch’s ANOVA: the test accounts for sample size and variance instead of using a single pooled error term in the same way as classic ANOVA.

R Chart 7: Group Size and Standard Deviation

The R group-size chart confirms the unequal sample sizes across studytime levels. It also shows that the group standard deviations are not identical.

This chart supports the recommendation to use Welch’s ANOVA when group sizes and variances deserve careful treatment.

R Chart 8: Welch’s ANOVA Summary Table

The R summary table confirms the final decision: Welch’s ANOVA rejects equal studytime group means.

The agreement across Python, R and SPSS makes the result strong for publication. The final report can state that the robust Welch test found significant mean differences across studytime groups.

SPSS, R, Python and Excel Workflows for Welch’s ANOVA

The same Welch’s ANOVA workflow can be reproduced in SPSS, R, Python and Excel. SPSS has a built-in Welch option under one-way ANOVA. R can run Welch’s one-way test with oneway.test(). Python can run Welch ANOVA using packages such as statsmodels or pingouin. Excel can support the descriptive and weighting steps, but SPSS, R or Python is better for final p-value reporting.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Welch’s ANOVA

SPSS Syntax for Welch’s ANOVA

* Welch's ANOVA in SPSS.
* Dependent variable: G3.
* Grouping factor: studytime.

TITLE "Welch's ANOVA: G3 by Studytime".

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

GRAPH
  /BAR(SIMPLE)=MEAN(G3) BY studytime
  /TITLE='Welch ANOVA: Mean G3 by Studytime'.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="welchs_anova_spss_output.pdf".

Python Code for Welch’s ANOVA

import pandas as pd
import numpy as np
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 statistics
summary = data.groupby("studytime")["G3"].agg(
    n="count",
    mean="mean",
    sd="std",
    variance="var"
).reset_index()

summary["welch_weight"] = summary["n"] / summary["variance"]
print(summary)

# Welch ANOVA using scipy's unequal-variance option if available
groups = [
    group["G3"].to_numpy()
    for _, group in data.groupby("studytime", observed=True)
]

try:
    result = stats.f_oneway(*groups, equal_var=False)
    print("Welch F:", result.statistic)
    print("Welch p:", result.pvalue)
except TypeError:
    print("Your SciPy version does not support equal_var=False in f_oneway.")
    print("Use pingouin.welch_anova(data=data, dv='G3', between='studytime') instead.")

# Optional pingouin workflow:
# import pingouin as pg
# welch_table = pg.welch_anova(data=data, dv="G3", between="studytime")
# print(welch_table)

R Code for Welch’s ANOVA

# Welch's ANOVA 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()

# Descriptive statistics
data %>%
  group_by(studytime) %>%
  summarise(
    n = n(),
    mean_G3 = mean(G3),
    sd_G3 = sd(G3),
    variance_G3 = var(G3),
    welch_weight = n / variance_G3,
    .groups = "drop"
  )

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

# Welch's one-way ANOVA
welch_result <- oneway.test(G3 ~ studytime, data = data, var.equal = FALSE)
print(welch_result)

# Classic ANOVA for comparison
classic_model <- aov(G3 ~ studytime, data = data)
summary(classic_model)

Excel Notes for Welch’s ANOVA

Excel support workflow:

1. Arrange the data:
   G3 | studytime

2. Create groups:
   Studytime 1, Studytime 2, Studytime 3, Studytime 4

3. Calculate group statistics:
   n = COUNT(group_range)
   mean = AVERAGE(group_range)
   SD = STDEV.S(group_range)
   variance = VAR.S(group_range)

4. Calculate Welch weights:
   weight = n / variance

5. Create charts:
   - group mean chart
   - boxplot by studytime
   - variance context chart
   - group size and SD chart

6. Formal Welch F and p-value:
   Use SPSS, R or Python for the final robust test result,
   adjusted degrees of freedom and p-value.

APA Reporting Wording

When reporting Welch’s ANOVA, include the test name, numerator degrees of freedom, adjusted denominator degrees of freedom, Welch F statistic, p-value and the group means. Also explain why Welch was used, especially if group sizes or variances are unequal.

Common Mistakes

When to Use Welch’s ANOVA

Use Welch’s ANOVA when you need to compare three or more independent group means and you are not comfortable assuming equal variances. It is also useful when the group sizes are unequal, especially when smaller groups have different variance patterns.

Compare this guide with Welch’s T Test, One Way ANOVA, Brown Forsythe ANOVA, ANOVA Assumptions, T Test vs ANOVA, T Test for Unequal Variances, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

Downloads and Resources for Welch’s ANOVA

Use these resources to reproduce the Welch’s ANOVA 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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