Quick Answer: ANOVA in R Result

The worked ANOVA in R example compares G3 final grade across four studytime groups. The group mean chart shows that average G3 is lowest for studytime group 1, higher for group 2, highest for group 3 and still high for group 4. The visible means are about 10.844, 12.092, 13.227 and 13.057.

The R F distribution chart shows an observed F statistic around 15.88, while the critical F value is around 2.62. The observed F statistic is far beyond the critical value, so the null hypothesis of equal group means is rejected. The effect size summary shows eta squared close to .069, omega squared and epsilon squared around .064, and Cohen’s f around .27. This gives a medium effect-size interpretation.

What Is ANOVA in R?

ANOVA in R means using R or RStudio to test whether three or more group means are statistically different. In a one-way ANOVA, one categorical factor is used to compare one numeric outcome. In this example, studytime is the categorical grouping factor and G3 final grade is the numeric outcome.

R can run ANOVA through aov() or through a linear model with lm(). For many beginner tutorials, aov(G3 ~ studytime, data = df) is the clearest command because it matches the ANOVA language directly. The lm() approach is also useful because ANOVA is a special case of the general linear model.

The R output in this guide is not only a table. It includes visual evidence for group mean differences, group spread, sum-of-squares decomposition, the F distribution, residual shape, residual normality and effect size. These charts help explain both the statistical decision and the practical size of the result.

Before using ANOVA in R, it helps to understand Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, P Value, and Null and Alternative Hypothesis.

ANOVA Formula Used in R

A one-way ANOVA model in R 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 studytime group effect, and ε_ij is the residual error. In R, this model is written as G3 ~ studytime.

ANOVA F Statistic

The F statistic compares between-group mean variation with within-group variation. In the R chart, the observed F statistic is about 15.88 and the critical F value is about 2.62. This means the studytime group means differ more than expected from random within-group variation.

Eta Squared Formula

Eta squared shows the proportion of total G3 variation explained by studytime. In this example, eta squared is about .069, so studytime explains about 6.9% of the total variation in final grades.

Null and Alternative Hypothesis for ANOVA in R

The R one-way ANOVA tests whether mean G3 is equal across all four studytime groups. The null hypothesis says the group means are equal. The alternative hypothesis says at least one group mean is different.

Dataset and R Variables Used

The worked example uses a student performance dataset. The outcome variable is G3 final grade, and the grouping variable is studytime. In R, G3 should be numeric and studytime should be treated as a factor before fitting the ANOVA model.

ANOVA in R should be interpreted with assumption checks. Helpful related guides include ANOVA Assumptions, ANOVA in Python, ANOVA Effect Size, Levene Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test, Outlier Detection, and Effect Size.

R Chart-by-Chart Interpretation

The R charts below show the complete ANOVA workflow: group means, group distributions, sum of squares, F statistic, residual histogram, residual Q-Q plot and effect size summary. These charts explain the direction, statistical decision, diagnostic pattern and practical size of the result.

R Chart 1: Group Means with 95% Confidence Intervals

This chart shows the average G3 score for each studytime group in the R ANOVA workflow. Studytime group 1 has the lowest mean, group 2 is higher, group 3 has the highest mean, and group 4 remains close to group 3. The chart shows a clear improvement in final grade as studytime moves from the lowest category to the higher categories.

The confidence intervals show uncertainty around the estimated group means. The first three studytime groups have comparatively tighter intervals, while group 4 has a wider interval. This wider interval shows that the group 4 mean is less precise, but its center is still in the higher G3 range.

In the final R ANOVA report, this chart should be used to describe the direction of the group difference. It shows which studytime groups have higher and lower G3 means before the F test is interpreted.

R Chart 2: Boxplot of G3 by Studytime Group

This boxplot shows the full distribution of G3 scores within each studytime group. The median is lower in group 1, higher in group 2, and highest across groups 3 and 4. This distribution pattern supports the same result shown by the group mean chart.

The boxplot also shows low outliers in the lower studytime groups, especially groups 1 and 2. These low-score cases explain why the residual charts show a lower-tail issue. Groups 3 and 4 are centered higher and do not show the same strong low-outlier pattern in this figure.

For ANOVA interpretation in R, this chart is important because it shows both group differences and within-group overlap. The groups differ in center, but the boxes and whiskers still overlap. This supports a medium effect-size conclusion rather than a claim that studytime completely separates student performance.

R Chart 3: Sum of Squares Decomposition

This chart separates the ANOVA variation into between-group and within-group components. The between-group sum of squares is the part connected to studytime group differences, while the within-group sum of squares is the part left inside the groups.

The within-group variation is much larger than the between-group variation. This is expected in student performance data because G3 is affected by many factors besides studytime. The between-group component is still large enough relative to the error term to produce a significant F statistic.

In reporting, this chart explains the ANOVA table. R calculates the F statistic by converting these sum-of-squares components into mean squares and then dividing the between-group mean square by the within-group mean square.

R Chart 4: F Statistic Distribution Curve

This chart shows the R ANOVA F distribution with the observed F statistic and the critical F value marked. The observed F statistic is about 15.88, while the critical F value is about 2.62.

The observed F statistic is far to the right of the critical value. This means the studytime group means differ more than expected from ordinary within-group variation. The result supports rejecting the null hypothesis that all studytime group means are equal.

In the R results section, this chart should be linked to the summary(aov_model) output. The report should state that the ANOVA was statistically significant, then use the effect size chart to explain how large the group effect is.

R Chart 5: Residual Histogram

This histogram shows the residual distribution from the R ANOVA model. Most residuals are centered around zero, which means the fitted group means capture the middle of the G3 data reasonably well.

The lower tail extends farther than the upper tail. This shows that some students had much lower G3 values than expected for their studytime group. This same pattern appears in the boxplot as low outliers and in the Q-Q plot as lower-tail departure.

In reporting, this chart supports a careful assumption statement. The residuals are mostly centered, but the lower tail is heavier than ideal. The report should not say that residual normality is perfect; it should say that the center is acceptable but tail departures are present.

R Chart 6: Residual Q-Q Plot

The Q-Q plot shows that the central residuals follow the diagonal line fairly well. This means the middle part of the residual distribution is reasonably close to normal.

The strongest departure is visible in the lower-left tail, where several residuals fall below the expected normal line. The upper tail also bends away from the line, but the lower-tail departure is the clearer diagnostic issue in this output.

In the R ANOVA report, this chart should be used to write a realistic residual-normality statement. The residuals are acceptable in the center, but tail departures should be acknowledged. Formal checks can include Shapiro-Wilk Test, Anderson-Darling Test, Jarque-Bera Test, Lilliefors Test and Kolmogorov-Smirnov Test.

R Chart 7: Effect Size Summary

This effect size summary gives the practical size of the studytime effect. Eta squared and partial eta squared are close to .069, while omega squared and epsilon squared are slightly lower, around .064. Cohen’s f is about .27.

The chart labels the effect as medium. This means the R ANOVA result is not only statistically significant but also meaningful enough to discuss as a practical difference. At the same time, the effect is not huge because most variation in G3 remains inside the groups.

In the final R report, this chart should be used after the F test. A strong report says that the studytime effect was significant and medium in size, explaining about 6.9% of the variation in G3.

R Workflow with aov, lm, summary and anova

The R workflow has five main parts: load the data, prepare the variables, run the ANOVA model, check assumptions and calculate effect size. The aov() function is the most direct beginner-friendly command, while lm() is helpful because it connects ANOVA to regression and the general linear model.

SPSS, R, Python and Excel Workflows for ANOVA

The same ANOVA model can be reproduced in SPSS, R, Python and Excel. This post focuses on R, but the comparison table helps readers understand how the same analysis is run across common software.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for ANOVA in R

R Code: Full One-Way ANOVA with aov()

# ANOVA in R: One-way ANOVA example
# Dependent variable: G3
# Grouping factor: studytime

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")])

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

print(group_summary)

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

# ANOVA table
summary(anova_model)

# Linear-model equivalent
lm_model <- lm(G3 ~ studytime, data = df_model)
anova(lm_model)

# Residual values
df_model$fitted <- fitted(anova_model)
df_model$residual <- residuals(anova_model)

head(df_model)

R Code: Assumption Checks

# Residual normality check
shapiro.test(residuals(anova_model))

# Base R diagnostic plots
par(mfrow = c(2, 2))
plot(anova_model)

# Levene test for homogeneity of variance
# install.packages("car")
library(car)
leveneTest(G3 ~ studytime, data = df_model)

# Brown-Forsythe style check using median-centered Levene
leveneTest(G3 ~ studytime, data = df_model, center = median)

# IQR outlier count by group
iqr_count <- function(x) {
  q1 <- quantile(x, .25, na.rm = TRUE)
  q3 <- quantile(x, .75, na.rm = TRUE)
  iqr <- q3 - q1
  lower <- q1 - 1.5 * iqr
  upper <- q3 + 1.5 * iqr
  sum(x < lower | x > upper, na.rm = TRUE)
}

aggregate(G3 ~ studytime, data = df_model, FUN = iqr_count)

R Code: Effect Size for ANOVA

# Manual ANOVA effect size calculation in R

anova_table <- summary(anova_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))

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

print(effect_sizes)

# Optional package method
# install.packages("effectsize")
library(effectsize)
eta_squared(anova_model)
omega_squared(anova_model)
epsilon_squared(anova_model)

Python Code for the Same ANOVA

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

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"])

model = ols("G3 ~ C(studytime)", data=df_model).fit()
anova_table = sm.stats.anova_lm(model, typ=2)

print(anova_table)
print(df_model.groupby("studytime")["G3"].agg(["count", "mean", "std", "var"]))

SPSS Syntax for the Same ANOVA

* One-way ANOVA equivalent to R model.
* Dependent variable: G3.
* Grouping factor: studytime.

TITLE "One-Way ANOVA: G3 by Studytime".

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

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

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="ANOVA-in-R-Equivalent-SPSS-Output.pdf".

Excel Formulas for ANOVA in R Results

Group mean:
=AVERAGE(group_range)

Group standard deviation:
=STDEV.S(group_range)

Total sum of squares:
=SS_Between + SS_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

ANOVA table:
Data > Data Analysis > ANOVA: Single Factor

APA Reporting Wording

When reporting ANOVA in R, include the dependent variable, grouping factor, group means, F statistic, degrees of freedom, p value, effect size and assumption checks. R output should be converted into clear reporting language instead of pasted as raw console output only.

Common Mistakes

When to Use ANOVA in R

Use ANOVA in R when you have one numeric outcome and one or more categorical group variables. A one-way ANOVA is used with one categorical factor. A two-way ANOVA is used with two categorical factors. ANCOVA is used when the group comparison includes a continuous covariate.

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

Downloads and Resources for ANOVA in R

Use these resources to reproduce the ANOVA in R workflow. Replace placeholder links with final hosted dataset, R script, Python validation script, SPSS syntax and Excel workbook URLs after uploading them to WordPress Media Library.

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