Effect Size Explained: Cohen’s d, r, Eta Squared and Cramér’s V
Reviewed by Prof. Dr. Alexander Holtermann · Updated 16 October 2026 · 6 min read
An effect size tells you how large a difference or relationship is, independent of sample size. Cohen’s d of 0.2, 0.5 and 0.8 is small, medium, large.
Key takeaways
- The p-value tells you whether an effect is likely to be more than chance; the effect size tells you how large it is. APA style requires both.
- Cohen’s d expresses a mean difference in standard deviation units:
d = (M₁ − M₂) / SD_pooled. Benchmarks: 0.2 small, 0.5 medium, 0.8 large. - For relationships use r (0.1 / 0.3 / 0.5), for ANOVA η² or partial η² (.01 / .06 / .14), for chi-square tests φ or Cramér’s V.
- Cohen’s benchmarks are a fallback. Interpret effect sizes in the context of your field and of previous studies whenever possible.
- Effect sizes are needed for power analysis and meta-analysis, which is why reporting them makes your study useful to others.
Why effect sizes matter
Imagine two studies of a new learning app. Study A with 40 students finds an improvement of 2 points, p = .21. Study B with 4,000 students finds an improvement of 0.3 points, p < .001. Which effect is more important? The p-values suggest Study B, but the improvement there is tiny. The p-value mixes effect size and sample size; the effect size isolates the magnitude.
Effect sizes are standardized: they do not depend on the unit of measurement or on n. That makes results comparable across studies with different scales. They are also the input for power analysis (how many participants do you need?) and for meta-analysis (what is the average effect across all studies?). APA 7 asks you to report an effect size for every main result.
The most important effect sizes at a glance
Different tests have different effect sizes. Cohen (1988) suggested the benchmark values in the table as rough guidance for the behavioral sciences.
| Effect size | Used with | Small | Medium | Large |
|---|---|---|---|---|
| Cohen’s d (or Hedges’ g) | t-test, two means | 0.20 | 0.50 | 0.80 |
| r (Pearson, or r from z or t) | correlation, Mann-Whitney U | .10 | .30 | .50 |
| η² / partial η² | ANOVA | .01 | .06 | .14 |
| Cohen’s f | ANOVA, power analysis | .10 | .25 | .40 |
| φ or Cramér’s V (df = 1) | chi-square test | .10 | .30 | .50 |
| R² | regression | .02 | .13 | .26 |
| Odds ratio | logistic regression, 2 × 2 tables | about 1.5 | about 2.5 | about 4.3 |
Cohen’s d: worked example
Eight students with a study plan score 14, 16, 15, 18, 17, 19, 16, 17 points (M = 16.50, SD = 1.60); eight without a plan score 13, 14, 12, 15, 16, 13, 14, 15 (M = 14.00, SD = 1.31).
Pooled standard deviation: SD_pooled = √(((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2)) = √((7 · 2.571 + 7 · 1.714) / 14) = 1.46
d = (16.50 − 14.00) / 1.46 = 1.71
- Interpretation: the groups differ by 1.71 standard deviations, a very large effect by Cohen’s standards.
- Overlap view: with d = 1.71 the average student with a plan scores higher than about 96% of students without one (if scores are normal). With d = 0.5 this share is about 69%.
- Hedges’ g corrects d for small samples:
g ≈ d · (1 − 3 / (4(n₁ + n₂) − 9))= 1.71 · (1 − 3 / 55) = 1.61. Use g with small groups or in meta-analyses. - Paired designs: d_z = mean difference / SD of differences. For a before/after anxiety study with a mean drop of 3.83 and SD of differences 1.60, d_z = 2.39. Say which variant you report, because d_z is usually larger than a d based on the raw SDs.
Effect sizes for correlation, ANOVA and chi-square
Each test family has its own natural effect size. These examples come from the worked examples in the related articles.
| Test | Result | Effect size | Interpretation |
|---|---|---|---|
| Pearson correlation, study hours and exam score (n = 10) | r = .84, p = .002 | r = .84, r² = .70 | Large; 70% shared variance |
| One-way ANOVA, three teaching methods (n = 15) | F(2, 12) = 9.50, p = .003 | η² = .61, ω² = .53 | Large |
| Chi-square, study group × passing (n = 100) | χ²(1) = 9.89, p = .002 | φ = .31 | Medium |
| Mann-Whitney U, anxiety ratings (n = 14) | U = 3, z = −2.78, p = .005 | r = 2.78 / √14 = .74 | Large |
Converting between effect sizes
Sometimes a paper only reports a test statistic, or you want to compare a d with an r. For two groups of equal size these conversions are standard:
r = d / √(d² + 4); for example d = 0.5 gives r = .24.d = 2r / √(1 − r²); for example r = .30 gives d = 0.63.r = √(t² / (t² + df)); for the study-plan example t = 3.42, df = 14 gives r = .67.d ≈ 2t / √dfis a quick approximation for equal groups: 2 · 3.42 / √14 = 1.83. The exact value from the raw data (1.71) is better; with equal groups it equalst · √(1/n₁ + 1/n₂).
Interpreting effect sizes sensibly
Cohen himself warned against using his benchmarks mechanically. They were meant as a last resort when nothing better is known. A “small” effect can be very important (a medication that reduces mortality by a small amount affects many lives), and a “large” effect in a small lab study may shrink in a replication.
- Compare your effect with effects reported in similar studies and meta-analyses.
- Translate it into practical terms: points on the exam, hours saved, percentage of students who pass.
- Report a confidence interval for the effect size if your software provides one; small samples give very wide intervals.
- Do not call an effect “absent” just because p > .05; a medium effect with a wide CI is simply uncertain.
How to report effect sizes in APA style
Put the effect size directly after the p-value, with the same care as the test statistic. Use two decimals. Values that cannot exceed 1 in absolute value (r, η², V) have no leading zero.
Frequently asked questions
What is a good effect size?
There is no universal answer. As a rough guide, Cohen’s d of 0.2 is small, 0.5 medium and 0.8 large; for r the values are .10, .30 and .50. Compare with similar studies in your field.
What is the difference between effect size and p-value?
The p-value shows whether a result is likely to be more than chance and depends heavily on sample size. The effect size shows how large the difference or relationship is and is independent of sample size.
Can Cohen’s d be larger than 1?
Yes. d = 1 means the means differ by one standard deviation. Values above 1 are common for strong interventions or when groups differ a lot.
Which effect size should I report for ANOVA?
η² or partial η² is most common; for one-way ANOVA both are identical. ω² is less biased in small samples. For pairwise follow-up comparisons, report Cohen’s d.
Do I need effect sizes for non-significant results?
Yes. Effect sizes show the magnitude of what you found regardless of significance and are needed for future power analyses and meta-analyses.
Sources and further reading
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.).
- Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.
Your university’s examination regulations and style guide always take precedence.
Related articles
- p-value and significanceThe p-value is the probability of a result at least as extreme as yours if the null hypothesis were true. p < .05 is usually called significant.
- t-testA t-test checks whether two means differ significantly: two independent groups, two paired measurements, or one mean against a fixed value.
- Type I and Type II errorsA Type I error is a false positive (rejecting a true null hypothesis, risk α); a Type II error is a false negative (missing a real effect, risk β).
- Sample sizeFor a survey with 95% confidence and a ±5% margin of error you need about 385 responses. For group comparisons, a power analysis sets the size.
- Reporting statistics in APAAPA style reports test statistic, degrees of freedom, exact p-value and effect size, e.g. t(13.46) = 3.42, p = .004, d = 1.71. Examples for every test.
- Confidence intervalA 95% confidence interval gives a range of plausible values for a population parameter. It shows both the size of an estimate and its precision.
