Eta-Squared as a Measure of Explained Variation

Why Measure the Size of Group Differences?

A one-way ANOVA assesses evidence about whether the population means of several groups are equal. Its F statistic and p-value address that inferential question, but they do not directly describe how much of the observed variation in the response is associated with group membership. Eta-squared provides a descriptive measure of that proportion.

A small p-value does not by itself tell us whether the observed group differences are large in a practical sense. Conversely, an effect-size estimate alone does not establish that population means differ. Eta-squared adds information about the size of the observed group separation, while the F test addresses a different question.

Definition: In a one-way ANOVA, eta-squared, written $\eta^2$, is the between-group sum of squares divided by the total sum of squares: $$ \eta^2 = \frac{SSB}{SST}. $$ It is the proportion of the total observed variation in the response that is associated with differences among the sample group means.

Calculating Eta-Squared

Suppose there are $k$ groups. Let $x_{ij}$ be the response for observation $j$ in group $i$, let $n_i$ be the size of group $i$, and let $n$ be the total number of observations. Write $\bar{x}_i$ for the sample mean in group $i$, and $\bar{x}$ for the overall sample mean. The sums of squares are

SSB = \sum_{i=1}^{k} n_i(\bar{x}_i-\bar{x})^2,\qquad SST = \sum_{i=1}^{k}\sum_{j=1}^{n_i}(x_{ij}-\bar{x})^2.

The between-group sum of squares, $SSB$, measures how far the group means are from the overall mean, weighted by group size. The total sum of squares, $SST$, measures how far all observations are from the overall mean. Both use squared deviations, so their units are the square of the response units; their ratio is unitless.

The one-way ANOVA variation decomposition is $SST=SSB+SSW$, where $SSW$ is the within-group sum of squares. Thus eta-squared can also be calculated from the within-group sum of squares:

\eta^2 = \frac{SSB}{SST} = \frac{SSB}{SSB+SSW} = 1-\frac{SSW}{SST}.

This identity gives the proportion interpretation directly. Dividing the decomposition by $SST$, when $SST>0$, gives

1=\frac{SSB}{SST}+\frac{SSW}{SST}.

The two ratios therefore split the total observed variation into the part between group means and the part within groups. Eta-squared is the between-group part of that split. It ranges from 0 to 1 when the total variation is positive. A value of 0 means the sample group means all equal the overall mean; a value of 1 means there is no within-group variation.

Worked Example: Calculating Eta-Squared From the Observations

Suppose a hypothetical exercise compares completion times, in minutes, for three groups of three participants each. The recorded times are Group A: 4, 5, 6; Group B: 6, 7, 8; and Group C: 8, 9, 10. These are numerical observations classified into the three stated groups, so the descriptive sums of squares and their ratio can be calculated directly.

The group means are $\bar{x}_A=5$, $\bar{x}_B=7$, and $\bar{x}_C=9$. The overall mean is $\bar{x}=63/9=7$ minutes. Therefore,

SSB=3(5-7)^2+3(7-7)^2+3(9-7)^2 =12+0+12=24\ \text{minutes}^2.

Within each group, the deviations from that group’s mean are $-1,0,1$, so each group contributes $1+0+1=2$ square minutes to $SSW$. Across three groups, $SSW=6$ square minutes. The decomposition gives $SST=24+6=30$ square minutes, and hence

\eta^2=\frac{SSB}{SST}=\frac{24}{30}=0.800.

For these nine hypothetical completion times, 80.0% of the total observed variation is associated with differences among the three sample group means. The remaining 20.0% is within-group variation. This is a description of these observed data; it does not establish that group membership caused the differences.

Unequal Group Sizes and ANOVA Summaries

The formula for $SSB$ includes $n_i$, the size of each group. As a result, a group mean based on more observations contributes more to the between-group sum of squares than an equally distant mean based on fewer observations. Do not calculate $SSB$ by simply adding the squared distances between the group means and the overall mean without weighting them by group size.

Worked Example: Unequal Group Sizes

In a hypothetical study of task scores, Group A has scores 2 and 4, Group B has scores 5, 7, and 9, and Group C has scores 8 and 10. All scores are numerical, and each observation is assigned to one of the three groups, so the sums of squares describe variation in these seven recorded scores without requiring an inferential model assumption.

The group means are 3, 7, and 9. The overall mean is $\bar{x}=45/7$, because the scores sum to 45. The between-group sum of squares is

\begin{aligned} SSB &=2\left(3-\frac{45}{7}\right)^2 +3\left(7-\frac{45}{7}\right)^2 +2\left(9-\frac{45}{7}\right)^2\\ &=2\left(\frac{-24}{7}\right)^2 +3\left(\frac{4}{7}\right)^2 +2\left(\frac{18}{7}\right)^2\\ &=\frac{1152+48+648}{49} =\frac{1848}{49} \approx 37.714. \end{aligned}

For Group A, the deviations from its mean 3 are $-1$ and $1$, contributing 2 to $SSW$. For Group B, the deviations from 7 are $-2,0,2$, contributing 8. Group C contributes 2 from deviations $-1$ and $1$. Thus $SSW=2+8+2=12$, and

SST=SSB+SSW\approx 37.714+12=49.714,\qquad \eta^2=\frac{37.714}{49.714}\approx 0.7586.

About 75.9% of the observed variation in these task scores is associated with differences among the three sample group means. The group sizes matter in the calculation: omitting them would give an incorrect $SSB$.

Worked Example: Calculating Eta-Squared From an ANOVA Summary

Suppose a hypothetical one-way ANOVA summary for ratings of a product, measured on a numerical scale, reports $SSB=18.6$ and $SSW=31.4$, both in squared rating units. The total sum of squares is

$$ SST=SSB+SSW=18.6+31.4=50.0. $$

Using the reported sums of squares,

\eta^2=\frac{18.6}{50.0}=0.372.

For the observed ratings, 37.2% of the total variation is between the sample group means, and 62.8% is within groups. This percentage describes the partition of variation; it does not identify which particular group means differ or explain why they differ.

What Eta-Squared Does and Does Not Tell You

Eta-squared is a descriptive effect-size measure for the observed data. It is not the F statistic and is not a p-value. The F statistic compares mean squares and is used for inference about population means under the one-way ANOVA model. A p-value measures how unusual an F statistic at least as large as the observed one would be under the null hypothesis and model assumptions. Eta-squared instead summarizes the relative size of the between-group sum of squares in the observed total variation.

A larger eta-squared means a larger share of the observed variation lies between the sample group means. It does not mean that the outcome changed by that percentage, that the group means differ by that percentage, or that group membership caused that percentage of the outcome. The response units have been squared in the sums of squares and cancel in the ratio, so eta-squared has no units.

The value is tied to the particular response, observations, and group definitions being analyzed. Two analyses with the same eta-squared need not have the same practical importance: the outcomes may have different meanings, and the percentage alone does not describe the size of a difference in the response’s original units. Use the group means and the study context alongside eta-squared when describing the observed pattern.

If $SST=0$, all observed responses are identical, so the ratio $SSB/SST$ is undefined. Eta-squared should not be reported as a calculated proportion in that case. When $SST>0$, the decomposition ensures $0\leq SSB\leq SST$, and therefore $0\leq\eta^2\leq1$.

Common Mistake

Incorrect reasoning: “The eta-squared is 0.37, so group membership caused 37% of the ratings.” This turns a descriptive association in the observed data into a causal claim. The ratio itself gives no evidence that the groups were assigned in a way that permits a causal conclusion.

Correction: Say that 37% of the observed total variation in ratings is between the sample group means, or that 37% is associated with group membership in these data. A causal interpretation requires an appropriate study design and justification beyond the eta-squared calculation.

Key takeaway: Calculate $\eta^2=SSB/SST$, using $SST=SSB+SSW$ when helpful. Interpret the result as the proportion of observed response variation associated with differences among the sample group means—not as a causal percentage or as a substitute for the F test and its p-value.

Check Your Understanding

Use the definitions and variation decomposition above to answer these questions.

An ANOVA summary reports $SSB=12$ and $SSW=28$. What is $SST$, and what is eta-squared?
In words, what does an eta-squared of 0.25 mean for the observed data?
Why must $SSB$ be weighted by group sizes when calculated from group means?
Can an eta-squared of 0.40 by itself show that group membership caused 40% of the response? Explain.
What happens to the eta-squared calculation if every observed response is identical?