Quick Answer: Balanced ANOVA Result

The worked Balanced ANOVA example compares G3 final grade across four studytime groups after equalizing the group sizes. The original dataset was unbalanced, with group sizes around 212, 305, 97 and 35. The balanced ANOVA sample uses 35 cases in each group, so the final balanced sample contains 140 cases.

The balanced group means are close to one another: 12.229 for studytime group 1, 13.143 for group 2, 12.886 for group 3 and 13.057 for group 4. The F distribution chart shows observed F = 1.11 and critical F = 2.67. Because the observed F statistic is below the critical value, the balanced ANOVA does not show a statistically significant group mean difference at the .05 level.

What Is Balanced ANOVA?

Balanced ANOVA is an ANOVA design where every group has the same sample size. In a one-way balanced ANOVA, each category of the grouping factor contributes an equal number of observations. This makes the design easier to interpret because each group has the same weight in the mean comparison.

In this example, the original studytime groups were not equal. Studytime group 2 had the largest number of observations, and group 4 had the smallest number of observations. The balanced ANOVA sample used the smallest group size as the target, so all four groups were reduced to 35 observations each.

A balanced ANOVA is useful when the analyst wants a clean equal-size comparison. It can reduce the dominance of very large groups, but it also discards data from larger groups when balancing is done by subsampling. That is why the balanced result should be interpreted as a sensitivity check or equal-size comparison, not automatically as a replacement for the full dataset result.

Balanced ANOVA is closely related to ANOVA Assumptions, ANOVA Effect Size, ANOVA in Python, ANOVA in R, ANOVA in SPSS, Effect Size, P Value, and Null and Alternative Hypothesis.

Balanced ANOVA Formula

A one-way balanced ANOVA model can be written as:

Here, Y_ij is the observed G3 score for student j in studytime group i, μ is the grand mean, τ_i is the group effect, and ε_ij is the residual error. The balanced design means that every group has the same number of observations.

Equal Group Size Condition

This condition is the defining feature of the example. The original group sizes were unequal, but the balanced sample uses 35 observations from each studytime group.

ANOVA F Statistic

The F statistic compares between-group mean variation with within-group variation. In the balanced output, the observed F value is 1.11, while the critical F value is 2.67. Since the observed value is lower than the critical boundary, the balanced ANOVA does not reject the null hypothesis at α = .05.

Eta Squared Formula

Eta squared measures the proportion of total variation explained by the group factor. In this balanced ANOVA, eta squared is .0239, meaning studytime explains only about 2.39% of the total G3 variation in the balanced sample.

Null and Alternative Hypothesis for Balanced ANOVA

The balanced ANOVA tests whether mean G3 is equal across the four equally sized studytime groups. The null hypothesis says that all balanced group means are equal. The alternative hypothesis says that at least one balanced group mean differs.

Dataset and Balanced Sampling Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade, and the grouping factor is studytime. The original data are unbalanced because the four studytime groups do not have equal sample sizes.

The balanced ANOVA sample uses the smallest original group size as the target sample size. Since studytime group 4 has 35 cases, the balanced workflow uses 35 cases from each studytime group. This creates a final balanced sample of 140 cases.

Balanced sampling should be interpreted together with Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Effect Size, and Statistical Power.

Python Chart-by-Chart Interpretation

The Python charts below show the balanced ANOVA workflow from sample balancing to final interpretation. They explain the original versus balanced group sizes, balanced group means, balanced boxplot, sum of squares, F distribution, residual histogram and effect size summary.

Python Chart 1: Original vs Balanced Group Sizes

This chart shows why a balanced ANOVA workflow was needed. The original group sizes are clearly unequal: studytime group 2 is the largest group, group 1 is also large, group 3 is smaller, and group 4 is the smallest. The balanced bars show that each group was reduced to the same sample size of 35 cases.

The chart is important because it shows that balancing is not a statistical test by itself. It is a sampling decision before the ANOVA is run. The equal orange bars mean that every studytime group receives the same weight in the balanced ANOVA comparison.

In the final report, this chart should be used to explain the difference between the full unbalanced dataset and the balanced analysis sample. The result should be described as a balanced-sample ANOVA, not as the full-data ANOVA.

Python Chart 2: Balanced Group Means with 95% Confidence Intervals

This chart shows the balanced mean G3 values after every studytime group was set to 35 cases. The means are 12.229 for group 1, 13.143 for group 2, 12.886 for group 3 and 13.057 for group 4.

The group means are close together. Group 2 has the highest mean, group 1 has the lowest mean, and groups 3 and 4 sit between or near the top. The difference between the lowest and highest balanced means is less than one G3 point.

This chart explains why the balanced ANOVA F statistic is small. The group means do not separate strongly after balancing, so the between-group variation is limited. The result is a much weaker group pattern than an analysis where large unequal groups dominate the comparison.

Python Chart 3: Balanced Boxplot of G3 by Studytime Group

This boxplot shows the distribution of G3 inside each balanced studytime group. The medians are similar across groups, with group 1 centered around 12 and the other groups centered around 13. The boxes overlap strongly, which supports the non-significant ANOVA decision.

Group 1 contains a few visible outliers, including low and high values. Group 3 has a low point near 8, and group 4 has a wider spread that reaches from about 6 to 19. The group centers are not far apart, but the within-group spread remains visible.

For balanced ANOVA reporting, this chart supports a practical conclusion: the balanced groups do not show a strong separation in G3 scores. The distributions overlap enough that the observed mean differences are small compared with within-group variation.

Python Chart 4: Sum of Squares Decomposition

This chart separates total variation into between-group and within-group components. The between-group sum of squares is very small, while the within-group sum of squares is much larger. This means most G3 variation remains inside the studytime groups rather than being explained by differences between studytime groups.

The small between-group bar matches the group mean chart. Since the four balanced means are close together, there is not much explained group variation. The large within-group bar shows that individual scores still vary substantially within each studytime group.

In the final article, this chart should be used to explain the F result. The observed F statistic is small because the between-group mean square is not large enough compared with the within-group mean square.

Python Chart 5: F Statistic Distribution Curve

This chart shows the balanced ANOVA F distribution. The observed F statistic is 1.11, while the critical F value is 2.67. The observed F line is to the left of the critical F line.

Because the observed F value does not reach the critical boundary, the balanced ANOVA does not reject the null hypothesis at the .05 level. The result means the balanced studytime group means are not different enough relative to within-group variation.

This chart should be used as the main decision visual. It shows the statistical result more clearly than a table alone: the group effect is present numerically, but it is not strong enough to be declared statistically significant in the balanced sample.

Python Chart 6: Balanced ANOVA Residual Histogram

This histogram shows the residuals from the balanced ANOVA model. Most residuals are clustered near zero, which means many observed G3 scores are reasonably close to their balanced group mean.

The residuals extend to about -7 on the negative side and about 6 on the positive side. The shape is not perfectly normal, but it is centered around zero and does not show a single extreme tail dominating the entire plot.

In reporting, this chart supports a moderate diagnostic statement. The balanced ANOVA residuals are centered near zero, but the distribution still shows real within-group variation. This is consistent with the large within-group sum of squares and the non-significant F result.

Python Chart 7: Balanced ANOVA Effect Size Summary

This effect size chart shows that eta squared and partial eta squared are both 0.0239. Omega squared and epsilon squared are both 0.0024. Cohen’s f is 0.1566.

The effect size values confirm that the balanced studytime effect is small. Eta squared means that studytime explains only about 2.39% of the total variation in G3 in the balanced sample. Omega squared and epsilon squared are even smaller because they adjust the estimate downward.

In the final report, this chart should be used after the F statistic. The correct interpretation is that the balanced ANOVA did not find a statistically significant group difference, and the effect size is small in practical terms.

R Chart-by-Chart Validation

The R validation charts repeat the same balanced ANOVA workflow in a second software environment. The R charts confirm the equal group sizes, similar group means, overlapping boxplots, small between-group variation, non-significant F decision, centered residuals and small effect-size values.

R Chart 1: Original vs Balanced Group Sizes

This R chart confirms the same sampling structure shown in Python. The original studytime groups are unequal, while the balanced sample uses the same number of observations from each group.

The equal balanced bars show that the ANOVA validation is based on 35 cases per group. This confirms that R is checking the same balanced design, not the original unbalanced dataset.

In reporting, this chart should be used to verify the analysis design. It shows that the balanced result is based on equal group representation and therefore answers a balanced-sample question.

R Chart 2: Balanced Group Means with 95% Confidence Intervals

This R chart labels the balanced group means as 12.229, 13.143, 12.886 and 13.057. The values match the Python chart and confirm that the balanced groups are close together.

The confidence intervals overlap visually. This supports the non-significant ANOVA decision because the group mean differences are not large compared with the uncertainty and within-group variation.

In the final report, this R chart confirms the direction and size of the balanced mean pattern. Group 2 is highest, group 1 is lowest, but the separation is not large enough to produce a significant balanced ANOVA result.

R Chart 3: Balanced Boxplot of G3 by Studytime Group

The R boxplot confirms that the balanced group distributions overlap strongly. The medians are close, and the boxes are not separated into clearly different score regions.

Group 4 shows a relatively wide range, while group 1 includes visible outlier points. These distribution features explain why within-group variation remains large even after balancing the group sizes.

In reporting, this chart supports the conclusion that balanced sampling did not create a strong group separation. The group means differ slightly, but individual G3 scores still vary considerably within each group.

R Chart 4: Sum of Squares Decomposition

This R chart confirms that within-group variation is much larger than between-group variation. The studytime group effect explains only a small part of the total G3 variation in the balanced sample.

The chart matches the small eta squared value. Since eta squared is calculated from the between-group sum of squares divided by the total sum of squares, a small between-group bar naturally produces a small explained-variance result.

In reporting, this chart validates the core ANOVA logic. The balanced F statistic is small because the explained group variation is weak compared with the unexplained within-group variation.

R Chart 5: F Statistic Distribution Curve

The R F distribution curve confirms the same decision as Python. The observed F statistic is 1.11, and the critical F value is 2.67.

The observed F statistic remains to the left of the critical F line. This confirms that the balanced ANOVA result is not statistically significant at the .05 level.

In reporting, this chart verifies that the non-significant decision is stable across software. Both Python and R show that the balanced group means do not differ enough to reject the null hypothesis.

R Chart 6: Balanced ANOVA Residual Histogram

The R residual histogram confirms that residuals are centered near zero. This means the balanced group means are located near the center of the observed scores for many cases.

The distribution still has values on both sides of zero, showing that the model leaves considerable within-group variation. This matches the large within-group sum of squares.

In reporting, this chart supports the same diagnostic statement as the Python histogram. The balanced ANOVA residuals are centered around zero, but the model does not explain most of the individual score variation.

R Chart 7: Balanced ANOVA Effect Size Summary

This R chart confirms the same effect-size values. Cohen’s f is 0.1566, eta squared and partial eta squared are 0.0239, and omega squared and epsilon squared are 0.0024.

The chart makes the practical result clear. The balanced studytime effect is small, and the adjusted effect-size measures are close to zero. This means the balanced sample does not support a strong practical group effect.

In the final article, this chart should be used as the practical interpretation summary. It confirms that the balanced ANOVA result is both statistically non-significant and small in explained variance.

SPSS Output Interpretation for Balanced ANOVA

The SPSS output PDF should be used as the final software verification file for the Balanced ANOVA workflow. It documents the balanced one-way ANOVA procedure, group comparison, equal-size sample structure, ANOVA table, assumption checks and reporting evidence.

Download Balanced ANOVA SPSS Output PDF

SPSS Output Items to Read

SPSS Reporting Summary

SPSS, R, Python and Excel Workflows for Balanced ANOVA

The same Balanced ANOVA workflow can be reproduced in SPSS, R, Python and Excel. The important step is not only running ANOVA. The important step is first creating or selecting a balanced sample where every group has the same number of observations.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Balanced ANOVA

SPSS Syntax for Balanced ANOVA

* Balanced ANOVA in SPSS.
* Dependent variable: G3.
* Grouping factor: studytime.
* Use a balanced sample with equal group sizes before running ANOVA.

TITLE "Balanced ANOVA: G3 by Studytime".

FREQUENCIES VARIABLES=studytime.

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

UNIANOVA G3 BY studytime
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /SAVE=PRED RESID
  /CRITERIA=ALPHA(.05)
  /DESIGN=studytime.

EXAMINE VARIABLES=RES_1
  /PLOT HISTOGRAM NPPLOT BOXPLOT
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Balanced-ANOVA-SPSS-Output.pdf".

Python Code for Balanced ANOVA

import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels.formula.api import ols
import scipy.stats as stats

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

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

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

# Balance to the smallest group size
min_n = df_model.groupby("studytime").size().min()

balanced = (
    df_model
    .groupby("studytime", group_keys=False)
    .sample(n=min_n, random_state=123)
    .copy()
)

print("Original group sizes")
print(df_model["studytime"].value_counts().sort_index())

print("Balanced group sizes")
print(balanced["studytime"].value_counts().sort_index())

# Balanced one-way ANOVA
model = ols("G3 ~ C(studytime)", data=balanced).fit()
anova_table = sm.stats.anova_lm(model, typ=2)
print(anova_table)

# Group means
group_summary = balanced.groupby("studytime")["G3"].agg(["count", "mean", "std", "var"])
print(group_summary)

# Effect sizes
ss_between = anova_table.loc["C(studytime)", "sum_sq"]
df_between = anova_table.loc["C(studytime)", "df"]
ss_within = anova_table.loc["Residual", "sum_sq"]
df_within = anova_table.loc["Residual", "df"]
ss_total = ss_between + ss_within
ms_within = ss_within / df_within

eta_squared = ss_between / ss_total
partial_eta_squared = ss_between / (ss_between + ss_within)
omega_squared = (ss_between - df_between * ms_within) / (ss_total + ms_within)
epsilon_squared = (ss_between - df_between * ms_within) / ss_total
cohen_f = np.sqrt(eta_squared / (1 - eta_squared))

print("Eta squared:", eta_squared)
print("Partial eta squared:", partial_eta_squared)
print("Omega squared:", omega_squared)
print("Epsilon squared:", epsilon_squared)
print("Cohen's f:", cohen_f)

# Residuals
balanced["fitted"] = model.fittedvalues
balanced["residual"] = model.resid

# Assumption support
groups = [
    group["G3"].dropna().values
    for name, group in balanced.groupby("studytime")
]

levene_stat, levene_p = stats.levene(*groups, center="median")
print("Levene statistic:", levene_stat)
print("Levene p:", levene_p)

R Code for Balanced ANOVA

library(dplyr)

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

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

df_model <- na.omit(df[, c("G3", "studytime")])

# Original group sizes
table(df_model$studytime)

# Balance to smallest group size
min_n <- min(table(df_model$studytime))

set.seed(123)
balanced <- df_model %>%
  group_by(studytime) %>%
  slice_sample(n = min_n) %>%
  ungroup()

# Balanced group sizes
table(balanced$studytime)

# Balanced group means
aggregate(G3 ~ studytime, data = balanced, FUN = function(x) {
  c(n = length(x), mean = mean(x), sd = sd(x), variance = var(x))
})

# Balanced ANOVA model
model <- aov(G3 ~ studytime, data = balanced)
summary(model)

# Effect size calculation
anova_table <- summary(model)[[1]]

ss_between <- anova_table["studytime", "Sum Sq"]
df_between <- anova_table["studytime", "Df"]
ss_within <- anova_table["Residuals", "Sum Sq"]
df_within <- anova_table["Residuals", "Df"]
ss_total <- ss_between + ss_within
ms_within <- ss_within / df_within

eta_squared <- ss_between / ss_total
partial_eta_squared <- ss_between / (ss_between + ss_within)
omega_squared <- (ss_between - df_between * ms_within) / (ss_total + ms_within)
epsilon_squared <- (ss_between - df_between * ms_within) / ss_total
cohen_f <- sqrt(eta_squared / (1 - eta_squared))

data.frame(
  eta_squared = eta_squared,
  partial_eta_squared = partial_eta_squared,
  omega_squared = omega_squared,
  epsilon_squared = epsilon_squared,
  cohen_f = cohen_f
)

# Residual diagnostics
par(mfrow = c(2, 2))
plot(model)

Excel Formulas for Balanced ANOVA

Step 1:
Separate data by studytime group.

Step 2:
Use the same number of rows from each group.

Balanced group count:
=COUNT(group_range)

Group mean:
=AVERAGE(group_range)

Group standard deviation:
=STDEV.S(group_range)

Run balanced ANOVA:
Data > Data Analysis > ANOVA: Single Factor

Total sum of squares:
=SS_Between + SS_Within

Mean square between:
=SS_Between / df_Between

Mean square within:
=SS_Within / df_Within

F statistic:
=MS_Between / MS_Within

Eta squared:
=SS_Between / SS_Total

Omega squared:
=(SS_Between - df_Between * MS_Within) / (SS_Total + MS_Within)

Cohen's f:
=SQRT(EtaSquared / (1 - EtaSquared))

Residual:
=Observed_G3 - Group_Mean

APA Reporting Wording

When reporting Balanced ANOVA, clearly say that the analysis used an equal-size sample. This is important because the result is not the same as a full unbalanced ANOVA result. Report the balanced group size, group means, F statistic, degrees of freedom, p value and effect size.

Common Mistakes

When to Use Balanced ANOVA

Use Balanced ANOVA when each group has the same sample size or when the research design intentionally requires equal group representation. It is common in controlled experiments, classroom examples, simulation studies and sensitivity checks where unequal group sizes may affect interpretation.

For related mean-comparison tutorials, see ANOVA in SPSS, ANOVA in Python, ANOVA in R, ANOVA Effect Size, ANOVA Assumptions, ANCOVA, T Test vs ANOVA, Two Sample T Test, and Welch’s T Test.

Downloads and Resources for Balanced ANOVA

Use these resources to reproduce the Balanced ANOVA workflow. The SPSS PDF verifies the balanced ANOVA output, while the script and workbook placeholders can be replaced after uploading final files to the WordPress Media Library.

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