Quick Answer: Eta Squared Result

The worked Eta Squared example measures the effect size of studytime on G3 final grade. The group mean chart shows that mean G3 is lowest in studytime group 1, higher in group 2, highest in group 3 and still high in group 4. The ANOVA F distribution chart shows observed F = 15.88 and critical F = 2.62, so the studytime group effect is statistically significant.

The eta squared variance explained chart shows that studytime explains about 6.9% of the total variation in G3, while about 93.1% remains unexplained or within-group variation. The effect-size comparison chart places eta squared and partial eta squared around 0.0688, with omega squared around 0.0643 and epsilon squared around 0.0644.

What Is Eta Squared?

Eta Squared is an effect-size measure used mainly with ANOVA. It tells what proportion of the total variation in the dependent variable is explained by the group factor. In this example, eta squared answers a direct question: how much of the variation in G3 final grade is explained by studytime group?

The output shows eta squared around 0.0688. This means studytime explains about 6.88% of the total G3 variation. The variance explained pie chart rounds this to 6.9%, with the remaining 93.1% shown as unexplained or within-group variation.

This is why eta squared is important in reporting. A p value can say that an ANOVA result is statistically significant, but it does not tell how large the effect is. Eta squared gives the practical size of the group effect.

Eta squared should be interpreted with Effect Size, ANOVA Effect Size, Cohen’s F Formula, ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, P Value and Null and Alternative Hypothesis.

Eta Squared Formula

Eta squared is calculated by dividing the between-group sum of squares by the total sum of squares.

The numerator is the variation explained by the group factor. The denominator is the total variation in the dependent variable. In this example, the group factor is studytime and the outcome is G3 final grade.

Eta Squared as Percent Variance Explained

When eta squared is 0.0688, multiplying by 100 gives about 6.88%. The pie chart rounds this result to 6.9% explained variance.

Partial Eta Squared

Partial eta squared is common in SPSS GLM output. In a simple one-way ANOVA, eta squared and partial eta squared are the same or nearly the same because there is only one effect in the model.

Connection with Cohen’s f

Eta squared can also be converted into Cohen’s f. In this example, eta squared around 0.0688 produces Cohen’s f around 0.2717, which is interpreted as a medium effect.

Statistical Decision and Effect-Size Interpretation

The ANOVA F statistic and eta squared answer different questions. The F statistic tests whether the studytime group means differ more than expected by random within-group variation. Eta squared tells how much of the total G3 variation is explained by the studytime grouping.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade, and the grouping factor is studytime. Eta squared is calculated from the ANOVA sum of squares after comparing G3 means across the four studytime groups.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Coefficient of Variation, Frequency Distribution, Five Number Summary, Interquartile Range, Range, Skewness and Kurtosis.

Python Chart-by-Chart Interpretation

The Python charts below show the first eta squared workflow. They explain the group mean pattern, sum of squares decomposition, variance explained, comparison with adjusted effect sizes, ANOVA F decision, group distribution and eta squared interpretation thresholds.

Python Chart 1: Group Means with 95% Confidence Intervals

This chart shows the mean G3 score for each studytime group. 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 visible pattern explains why the ANOVA effect exists before eta squared is calculated.

The confidence intervals show uncertainty around each group mean. The first three groups have tighter intervals, while group 4 has a wider interval. The wider interval means group 4 is less precisely estimated, but its mean still remains in the higher grade range.

This chart gives the practical direction of the effect. Eta squared later converts this visible group pattern into a percent of total variation explained by studytime.

Python Chart 2: Sum of Squares Decomposition

This chart separates total variation into between-group and within-group components. The between-group part is the variation explained by studytime, while the within-group part is the variation left inside the studytime groups.

The within-group sum of squares is much larger than the between-group sum of squares. This means studytime explains a real part of the G3 variation, but most of the variation still comes from differences among students within the same studytime groups.

This chart is the direct base of eta squared. Eta squared is calculated by dividing the between-group sum of squares by the total sum of squares, so the relatively smaller explained bar produces an eta squared value around 0.0688.

Python Chart 3: Eta Squared Variance Explained Pie Chart

This pie chart converts eta squared into an easier variance-explained statement. Studytime explains 6.9% of G3 variation, while 93.1% remains unexplained or within-group variation.

The smaller explained slice does not mean the result is unimportant. Educational performance is affected by many factors, so one group factor explaining nearly seven percent of final-grade variation is meaningful.

This chart is the clearest practical interpretation of eta squared. It shows that studytime has a measurable effect, but it does not explain most of the differences among students.

Python Chart 4: Effect Size Comparison

This chart compares eta squared with related ANOVA effect-size measures. Eta squared and partial eta squared are about 0.0688. Omega squared is about 0.0643, and epsilon squared is about 0.0644.

The adjusted measures are slightly smaller than eta squared. This is expected because omega squared and epsilon squared reduce the estimate to avoid overstating the population effect.

The important result is that all values stay in the same practical range. The effect-size conclusion does not change when adjusted measures are used; studytime still has a meaningful medium-sized effect on G3.

Python Chart 5: ANOVA F Distribution Curve

This chart shows the ANOVA F distribution for the studytime effect on G3. The critical F value is 2.62, and the observed F statistic is 15.88.

The observed F statistic is far to the right of the critical F line. This means the studytime group means differ statistically, so it is meaningful to report an ANOVA effect size.

This chart separates significance from effect size. The F curve gives the decision that the group effect exists, while eta squared explains how much total variation is explained by that effect.

Python Chart 6: Boxplot of G3 by Studytime Group

This boxplot shows the full G3 distribution inside each studytime group. The lower studytime group has a lower center, while the higher studytime groups have higher centers. This supports the same direction shown in the group mean chart.

The groups still overlap, and the lower studytime groups contain low outliers. This explains why eta squared is not large. Studytime shifts the group centers upward, but individual G3 scores still vary widely inside each group.

For reporting, the boxplot gives the practical context behind eta squared. An eta squared value around 0.0688 fits a situation where group means differ, but the distributions are not completely separated.

Python Chart 7: Eta Squared Interpretation Thresholds

This chart compares the observed eta squared value with common interpretation thresholds. The observed value is around 0.069, which is above the small range and close to the medium interpretation zone shown in the chart.

The threshold chart supports a medium-effect conclusion. The value is meaningful enough to report as a practical effect, but it is not high enough to be described as a large effect.

This chart should be used as the final practical interpretation visual. It turns eta squared from a decimal value into a clear reporting category for readers.

R Chart-by-Chart Validation

The R validation charts repeat the same eta squared workflow in a second software environment. They confirm the same group mean direction, sum-of-squares pattern, variance explained percentage, effect-size comparison, ANOVA F decision, group distribution and threshold interpretation.

R Chart 1: Group Means with 95% Confidence Intervals

This R chart confirms the same group mean pattern as Python. Studytime group 1 is lowest, group 2 is higher, group 3 is highest and group 4 remains close to group 3.

The confidence interval pattern is also consistent. Group 4 has wider uncertainty, while the larger groups have narrower intervals. The direction of the effect remains stable across software.

In reporting, this chart confirms that eta squared is based on a real group mean pattern rather than an isolated calculation.

R Chart 2: Sum of Squares Decomposition

The R sum of squares chart confirms that within-group variation is much larger than between-group variation. This matches the Python chart and explains why eta squared is meaningful but not large.

The between-group component represents the studytime effect. The within-group component represents variation among students inside the same studytime categories.

In reporting, this chart verifies the formula logic. Eta squared is the explained portion divided by the total portion, so the smaller explained component produces an effect size near 0.069.

R Chart 3: Eta Squared Variance Explained Pie Chart

This R pie chart confirms the same variance-explained result. Studytime explains about 6.9% of G3 variation, while about 93.1% remains unexplained or within-group variation.

The chart makes the eta squared result easy to communicate. Eta squared is not only a decimal; it is the explained share of total outcome variation.

In the final article, this chart supports the statement that studytime has a measurable but not dominant effect on G3 final grade.

R Chart 4: Effect Size Comparison

This chart confirms that eta squared and partial eta squared are the larger values, while omega squared and epsilon squared are slightly smaller. The adjusted measures reduce the effect estimate modestly.

The practical conclusion remains the same across all effect-size measures. The studytime effect is meaningful and sits around the medium interpretation range.

In reporting, this chart supports a careful effect-size statement. Eta squared can be reported as the main value, while adjusted measures can be mentioned as a conservative check.

R Chart 5: ANOVA F Distribution Curve

The R F distribution curve confirms the same ANOVA significance decision. The observed F statistic is about 15.88, while the critical F value is about 2.62.

The observed F statistic is far beyond the critical value. This confirms that the studytime group means differ statistically in the ANOVA model.

In reporting, this chart should be paired with eta squared. The F statistic confirms the presence of the group effect, while eta squared explains the practical size of that effect.

R Chart 6: Boxplot of G3 by Studytime Group

The R boxplot confirms the same distribution pattern. The higher studytime groups have higher centers, but the distributions still overlap.

The overlap explains why eta squared is not large. A large eta squared would usually appear with much stronger separation among the group distributions.

In the final post, this chart validates the practical interpretation. The studytime effect is visible and meaningful, but most student-level variation remains inside the groups.

R Chart 7: Eta Squared Interpretation Thresholds

This R chart confirms the same threshold conclusion. The observed eta squared value is around 0.069, which places the effect in the medium interpretation area shown by the chart.

The observed value is not close to a large effect region, so the result should not be overstated. The correct interpretation is that studytime has a medium practical effect on G3.

In reporting, this chart gives the final effect-size category and helps readers understand what the eta squared decimal means.

SPSS Output Interpretation for Eta Squared

The SPSS output PDF is included as the downloadable software output for the Eta Squared workflow. SPSS commonly reports partial eta squared in GLM output. Eta squared can also be calculated from the ANOVA sum of squares by dividing between-group sum of squares by total sum of squares.

Download Eta Squared SPSS Output PDF

SPSS Output Items to Read

SPSS Reporting Summary

SPSS, R, Python and Excel Workflows for Eta Squared

The same Eta Squared workflow can be reproduced in SPSS, R, Python and Excel. The main steps are to run ANOVA, extract the sum of squares, calculate eta squared, and interpret the result as variance explained.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Eta Squared

SPSS Syntax for Eta Squared Workflow

* Eta Squared workflow in SPSS.
* Dependent variable: G3.
* Factor: studytime.

TITLE "Eta Squared for ANOVA Effect Size: 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.

* Eta squared can be calculated from the ANOVA table:
* eta squared = SS_between / SS_total.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Eta-Squared-SPSS-Output.pdf".

Python Code for Eta Squared

import pandas as pd
import numpy as np
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"]).copy()

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

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
cohens_f = np.sqrt(eta_squared / (1 - eta_squared))

print(anova_table)
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:", cohens_f)
print("Percent variance explained:", eta_squared * 100)

R Code for Eta Squared

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")])

model <- aov(G3 ~ studytime, data = df_model)
summary(model)

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
cohens_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,
  cohens_f = cohens_f,
  percent_explained = eta_squared * 100
)

Excel Formulas for Eta Squared

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

Total sum of squares:
=SS_Between + SS_Within

Eta squared:
=SS_Between / SS_Total

Partial eta squared:
=SS_Between / (SS_Between + SS_Within)

Percent variance explained:
=EtaSquared * 100

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

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

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

APA Reporting Wording

When reporting Eta Squared, include the ANOVA result first and then report eta squared as the practical effect-size measure. The effect size should not be reported without explaining which factor and outcome were used.

Common Mistakes

When to Use Eta Squared

Use eta squared when you need to explain how much total variation in an outcome is explained by an ANOVA factor. It is especially useful in one-way ANOVA tutorials, educational research, psychology examples, business experiments and student performance analysis.

For related guides, see ANOVA Effect Size, Cohen’s F Formula, ANOVA in SPSS, ANOVA in Python, ANOVA in R, ANOVA Assumptions, Balanced ANOVA, Brown Forsythe ANOVA, Brown-Forsythe Test, ANCOVA and Statistical Power.

Downloads and Resources for Eta Squared

Use these resources to reproduce the Eta Squared workflow. The SPSS output PDF is included as the software output file, while the script and workbook placeholders can be replaced with final uploaded files after they are added to the WordPress Media Library.

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