Darker shading means a larger effect size; the gold marker shows where your current |d| falls.
Cohen's d reference table
Click any row to load it into the calculator above.
| Cohen's d | r | R² | Effect size |
|---|
- Reporting effect sizes in psychology/social science papers
- Pre-processing for meta-analysis
- Sanity-checking effect sizes during peer review
Why effect size matters more than significance
A p-value answers a narrow question: if there were no effect at all, how surprising would this data be? It does not tell you how big the effect is. Because p-values shrink as samples grow, a large enough study will return p < 0.001 for a difference nobody would care about. An effect size is the part of the result that survives that problem — it describes the magnitude of the difference independently of how many people you measured.
This is why journals in psychology, medicine and education now expect effect sizes to be reported alongside p-values as a matter of course, and why the American Psychological Association's style guide has required them since its fifth edition.
The three measures on this page
- Cohen's d — the difference between two group means, expressed in standard deviations. A d of 0.5 means the groups differ by half a standard deviation. It has no upper bound.
- Correlation r — the strength and direction of a linear relationship between two continuous variables, running from −1 through 0 to +1.
- R² — the proportion of variance in one variable predictable from the other. It is simply r squared, so it is always positive and loses the direction of the relationship.
The constant 4 in the first formula is a = (n₁ + n₂)² ÷ (n₁ × n₂), which equals exactly 4 when the two groups are the same size. This tool assumes roughly equal group sizes. If your groups are badly unbalanced — say 20 against 200 — the d-to-r conversion here will be off, and you should compute a from your actual group sizes.
Cohen's thresholds, and why they are weaker than they look
| Cohen's d | r | R² | Conventional label |
|---|---|---|---|
| < 0.2 | < 0.10 | < 1% | Negligible |
| 0.2 – 0.5 | 0.10 – 0.24 | 1% – 6% | Small |
| 0.5 – 0.8 | 0.24 – 0.37 | 6% – 14% | Medium |
| > 0.8 | > 0.37 | > 14% | Large |
Cohen was explicit that these cut-offs were arbitrary conventions offered "for lack of a better basis", to be used only when a field has no benchmark of its own. Treating them as objective thresholds is the most common misuse of his work. In fields where interventions are cheap and applied at scale, a d of 0.1 can be enormously valuable; in clinical trials of an expensive drug with side effects, a d of 0.4 may not justify the treatment.
Worked example
A study reports that a tutoring programme raised test scores with Cohen's d = 0.5 — a "medium" effect by convention.
Converting: r = 0.5 ÷ √(0.25 + 4) = 0.5 ÷ 2.06 = 0.243, and R² = 0.243² = 0.059. So the programme accounts for about 5.9% of the variance in scores.
Both framings are accurate, and they land very differently on a reader. "A medium effect" sounds substantial; "explains 6% of the variance" sounds modest. This is exactly why reporting the effect size in more than one form is good practice.
Common mistakes
- Reading R² as "percentage of cases explained". It is the percentage of variance, which is a statement about squared deviations, not about individuals.
- Comparing d across studies that used different SDs. Cohen's d is scaled by the pooled standard deviation, so a study on a homogeneous sample will report a larger d for the same raw difference than one on a diverse sample. This is a well-documented problem in meta-analysis.
- Using Cohen's d for paired data without adjustment. The paired-samples case needs either the standard deviation of the differences or a correction for the correlation between measurements; using the raw pooled SD overstates the effect.
- Ignoring the confidence interval. A d of 0.8 from twelve participants carries an interval so wide it is compatible with almost anything. Effect sizes from small samples are as uncertain as everything else from small samples.
If you have raw data instead
This page converts between effect size measures you already have. If you are starting from raw numbers, run the comparison on the hypothesis test page first — the two-sample t-test there reports Cohen's d directly. To work out what sample size a given effect would need, use the power analysis tool.
Sources
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum. (Origin of the d thresholds and the d↔r conversion.)
- Borenstein, M., Hedges, L. V., Higgins, J. P. T. & Rothstein, H. R. (2009). Introduction to Meta-Analysis. Wiley. (On converting between effect size metrics.)
- Lakens, D. (2013). Calculating and reporting effect sizes. Frontiers in Psychology, 4, 863.