The p-value tells you whether an effect is statistically significant; the effect size tells you whether that effect matters. With large samples, almost any trivial difference produces a p below .05, and with small samples a large effect may not reach significance. That is why effect size is not optional: your committee will ask for it, indexed journals require it, and without it your results are incomplete. This guide covers the most common indices in psychology and health sciences, when to use each, and how to interpret them.
What effect size measures
Effect size quantifies the magnitude of a relationship or difference in standardised units, independently of sample size. Two studies with the same true effect will produce similar effect sizes even if one has 50 participants and the other 5,000; in contrast, they will have very different p-values. That is what makes effect size essential for clinical interpretability and for meta-analyses, which cannot combine p-values but can combine d or r.
Cohen's d: for comparing two means
Cohen's d is the most common index for group comparisons. It is calculated as the difference between the two means divided by the pooled standard deviation: a d of 0.5 means the groups are half a standard deviation apart. Cohen's (1988) reference values are the most widely used:
| Magnitude | Cohen's d |
|---|---|
| Small | 0.20 |
| Medium | 0.50 |
| Large | 0.80 |
Use these as guidance, not as universal rules. A "small" effect in pharmacology can be clinically important at the population scale, and a "large" effect in laboratory experimental psychology may be irrelevant in a clinical context. The serious recommendation is to anchor your interpretation in the literature of your specific field, not just in Cohen's table.
Common variants: Hedges' g is a correction of d for small samples (d overestimates the effect in small samples); with large samples the difference is negligible. When groups have very different sizes, use the weighted pooled standard deviation. The effect size calculator computes d, g and their confidence intervals from means, standard deviations and sample sizes.
Pearson r: for correlations
Pearson r as an effect size ranges from -1 to +1. Its reference values are also from Cohen:
| Magnitude | |r| |
|---|---|
| Small | 0.10 |
| Medium | 0.30 |
| Large | 0.50 |
Report r with its 95% confidence interval (obtained by Fisher's z transformation), not just the point estimate. Pearson r can also be converted to d when you need to compare studies that used different designs: d = 2r / sqrt(1 - r²).
eta² and partial eta²: for ANOVA
Eta squared (eta²) is the proportion of total variance explained by a factor in ANOVA. Partial eta squared (eta²p) is the proportion explained by that factor out of the total minus the variance of other factors, and it is what SPSS reports by default. They are different and their interpretation changes:
| Magnitude | eta² (Cohen) |
|---|---|
| Small | 0.01 |
| Medium | 0.06 |
| Large | 0.14 |
A known limitation of eta²: it overestimates the effect in the sample because the variance in the denominator includes measurement error. Omega squared (omega²) is the less-biased estimate of the population effect; use the omega calculator for that. In between-subjects designs with few groups and moderate samples, the difference between eta² and omega² can be substantial.
Cohen's f: for multi-group comparisons
Cohen's f is the square root of the ratio between the variance of group means and the within-group variance. It is used primarily to calculate a priori sample size in G*Power for ANOVA designs. Reference values:
| Magnitude | Cohen's f |
|---|---|
| Small | 0.10 |
| Medium | 0.25 |
| Large | 0.40 |
The relationship between f and eta² is f = sqrt(eta² / (1 - eta²)), so if you have one you can calculate the other. In practice, when calculating sample size for ANOVA in G*Power you enter f, even if you obtained eta² from a prior study; convert first.
OR, RR and NNT: for dichotomous outcomes in health research
In clinical studies with dichotomous outcomes (cure, relapse, mortality) the indices are different: the odds ratio (OR), the relative risk (RR) and the NNT (number needed to treat). The article on OR and RR in clinical research covers their interpretation and when to use each.
How to report effect size in APA 7
The standard format is to report the index with two decimal places immediately after the test statistic and the p-value, adding the confidence interval if available:
t(58) = 3.14, p = .003, d = 0.81, 95% CI [0.29, 1.32]
F(2, 87) = 7.43, p = .001, eta²p = .146
Always identify the index you are using (Cohen's d, Hedges' g, partial eta squared) because the names are used loosely in the literature. If you use SPSS's eta² but label it that way, a reviewer may ask for omega². More detail on how to report results by analysis type in the guides for ANOVA in APA 7 and correlation.
Effect size and sample size: the relationship that determines your design
The expected effect size is the most influential ingredient in sample size calculation. To detect a small effect (d = 0.2) with power .80 and alpha .05 you need more than 390 participants per group; for a large effect (d = 0.8) with the same parameters, 26 suffice. Choosing the expected effect well (from the literature, from a meta-analysis, from a pilot) is not a technical detail: it is the decision that determines whether your study has the power to detect what you are looking for. The article on sample size calculation with G*Power explains how to use it step by step.
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What is Cohen's d and when is it used?
Cohen's d is an effect size index for comparing two means. It is calculated as the difference between the means divided by the pooled standard deviation. Use it when you have a t-test or a two-group comparison and need to report not just whether the difference is significant, but how large it is.
What is a large, medium and small effect size according to Cohen?
Cohen's (1988) reference values for d are: small = 0.2, medium = 0.5, large = 0.8. For Pearson r: small = 0.1, medium = 0.3, large = 0.5. For eta squared: small = 0.01, medium = 0.06, large = 0.14. These are general benchmarks; what counts as large or small depends on the field of research.
What is the difference between eta squared and partial eta squared?
Eta squared is the proportion of total variance explained by the factor. Partial eta squared is the proportion explained by that factor out of the total minus the variance of other factors in the model. SPSS reports partial eta squared by default in ANOVAs. They are different and should not be interpreted the same way; partial eta squared is typically larger.
Is it mandatory to report effect size in a thesis?
Yes. APA 7 standards require reporting effect size alongside the test statistic and p-value. Indexed journals also require it, and thesis committees ask for it with increasing frequency. Without effect size, the results are incomplete.