Convert between Pearson r, r², Cohen's d, Hedges' g, odds ratio, log(OR), Fisher's z, and η². All formulas are shown alongside each conversion so you can verify and cite the exact transformation used.
Select the metric you have and enter its value. All other effect size measures will be computed using standard conversion formulas.
d = 2r / √(1 − r²)
r = d / √(d² + 4)
OR = exp(d × π / √3)
d = ln(OR) × √3 / π
r² = r × r
r = √(r²) [sign ambiguous]
Fisher's z = 0.5 × ln((1+r)/(1−r))
η² ≈ r² (bivariate case)
Example: Enter r = 0.3 to see d ≈ 0.6305, OR ≈ 3.1755, Fisher's z ≈ 0.3095, r² = 0.09, and η² ≈ 0.09.
Choose the effect size metric you have: r, r², d, OR, or log(OR).
Type the numeric value. The tool validates the input range automatically.
View converted values for all eight metrics with interpretations and the formula used.
Click Copy to copy all converted values to your clipboard for pasting into a report.
Most formulas assume equal group sizes and normally or logistically distributed outcomes. The r-to-d conversion d = 2r / √(1 – r²) is exact only for a point-biserial r with equal n per group.
It is better to extract the effect size that matches your study design directly (e.g., OR from a 2×2 table) rather than converting from a different metric. Conversions introduce additional assumptions.
The OR ≈ exp(dπ/√3) formula uses the logistic approximation. If you suspect non-logistic distributions, consider the probit approximation instead. Always state which formula you used.
When reporting converted effect sizes in a meta-analysis, best practice is to report both the original metric and the converted value, along with the formula used, so readers can assess the conversion’s validity.
A correlation to Cohen's d converter bridges the gap between two families of effect size metrics that researchers report across disciplines. Pearson's r quantifies the linear association between two continuous variables, whereas Cohen's d expresses the standardised mean difference between two groups. Cohen (1988) established the canonical benchmarks for interpreting d—0.2 (small), 0.5 (medium), and 0.8 (large)—and Rosenthal (1991) later demonstrated that every standardised mean difference can be re-expressed as a correlation and vice versa. The algebraic relationship d = 2r / √(1 − r²) assumes equal group sizes and a point-biserial correlation; when these assumptions hold, the conversion is exact. For unequal groups, Peterson & Brown (2005) proposed an approximation that adjusts the r-to-d transformation using the group size ratio, improving accuracy when the design is unbalanced. Our effect size calculator computes d, g, and r directly from raw summary data, whereas this converter transforms one already-computed metric into another so that heterogeneous study reports can enter the same pooled analysis.
An effect size conversion tool is indispensable during the data-extraction phase of any meta-analysis because primary studies seldom report the same statistic. Borenstein et al. (2009), in Introduction to Meta-Analysis, devote an entire chapter to conversion formulae and warn that each transformation carries distributional assumptions—the logistic approximation OR = exp(d × π / √3), for example, presupposes a logistic rather than normal error distribution. Reviewers should also consider attenuation correction (Spearman, 1904) when correlations are attenuated by measurement error in either variable, as disattenuating r before conversion can substantially alter the resulting d. Software such as Comprehensive Meta-Analysis (CMA) and the metafor package in R automate many of these conversions with built-in corrections, reducing the risk of manual computation errors during the extraction phase. The Cochrane Handbook (Higgins et al., 2023, Chapter 6) reinforces this point, advising reviewers to report both the original and the converted metric so that readers can evaluate the plausibility of the approximation. When only a p-value and a point estimate survive from the original study, our p-value to confidence interval converter can reconstruct the 95 % CI before you proceed with further conversions, ensuring that precision information is not lost in the extraction pipeline.
The r to d calculator implemented here also outputs Fisher's z, r², η², log(OR), and Hedges' g, covering the full spectrum of metrics required by PRISMA 2020 (Page et al., 2021) compliant reviews. Fisher's z transformation normalises the sampling distribution of r, which becomes increasingly skewed as the true correlation approaches ±1; meta-analysts therefore pool correlations on the z scale and back-transform afterward. In network meta-analysis, maintaining a consistent effect size scale across all pairwise comparisons is essential for valid indirect estimates, making reliable conversion tools a prerequisite for any multi-treatment synthesis. When partial correlations are reported rather than zero-order correlations, reviewers may also need to convert from partial to semi-partial correlation before deriving a standardised mean difference. When variability data are reported as standard errors rather than standard deviations, our SE to SD converter resolves the discrepancy before effect sizes are calculated, maintaining a consistent analytic pathway from raw extraction to final synthesis. Rosenthal (1991) showed that converting between r and d preserves the rank ordering of study effects, which is critical for sensitivity analyses that examine whether conclusions change when different metrics are pooled.
Methodological best practice dictates that reviewers should extract the native effect size whenever possible and convert only when the original metric is unavailable. Borenstein et al. (2009) further recommend documenting every conversion in a supplementary table so that the analytic chain remains fully transparent. When studies report medians and interquartile ranges instead of means and standard deviations, an intermediate step through our median/IQR to mean/SD estimator is necessary before any effect size can be derived. Once all studies share a common metric, the converted values feed into a forest plot that visualises the pooled estimate alongside individual study contributions—the final deliverable of a rigorous quantitative synthesis. By combining validated conversion formulae with complete formula transparency, this tool supports the reproducibility standards that the Cochrane Handbook and PRISMA 2020 demand of every modern systematic review.
The formula is d = 2r / √(1 – r²). This assumes equal group sizes and a point-biserial correlation. For unequal groups, a correction factor involving the group sizes should be applied. The conversion is exact when r represents the correlation between a dichotomous grouping variable and a continuous outcome.
Yes, r = √(r²). However, r-squared discards the sign of the original correlation. If you only have r², you cannot determine whether the original r was positive or negative without additional context. This calculator assumes a positive r when converting from r².
The logistic approximation OR = exp(d × π / √3) works well when the outcome follows a logistic distribution. For normally distributed outcomes, the probit approximation OR = exp(d × 1.65) may be more appropriate. The logistic approximation is standard in meta-analysis and is what this calculator uses.
Fisher’s z transformation is used in meta-analysis when pooling correlations. The sampling distribution of r is skewed (especially for large |r|), whereas Fisher’s z is approximately normal with a known standard error of 1/√(n–3). Always transform to Fisher’s z before pooling, then back-transform the pooled z to r.
For a bivariate relationship, eta-squared equals r-squared. Eta-squared represents the proportion of variance in the dependent variable explained by the independent variable. In ANOVA contexts, η² = SS_between / SS_total. The equivalence η² ≈ r² holds only in the two-group, one-predictor case.
Use the formula d = 2r / √(1 – r²). This assumes the correlation is point-biserial (between a binary grouping variable and a continuous outcome) with equal group sizes. For unequal groups, use d = 2r√(1 – r²) / p(1–p), where p is the proportion in one group. The standard formula is widely used in meta-analysis for combining correlation and SMD studies.
Fisher’s z transforms Pearson r into z = 0.5 × ln((1+r)/(1–r)), which has an approximately normal sampling distribution regardless of the true correlation. This transformation is essential when meta-analyzing correlations: pool the z-transformed values using inverse-variance weighting, then back-transform the pooled z to r. Without this step, the CI for r would be asymmetric and potentially exceed [–1, 1].
Yes. First convert the odds ratio to Cohen’s d using d = ln(OR) × √3 / π, then convert d to r using r = d / √(d² + 4). This two-step process (Borenstein et al., 2009) allows combining odds ratios with correlations in a single meta-analysis by converting all effect sizes to a common metric.
Our statisticians can handle effect size extraction, conversion, heterogeneity assessment, and publication bias testing for your entire systematic review.