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Effect Size Converter

Effect sizes (Cohen's d, r, odds ratio, eta-squared) put the magnitude of a result in a unit-free form so you can compare across studies and judge practical importance. Convert between any pair, get the small / medium / large label, and translate to practical metrics like CLES (probability one beats the other) and NNT.

i New to effect sizes? Read the 4-min primer ▾

What effect sizes are. An effect size answers “how big is the effect?” in a unit-free way. p-values say whether an effect exists; effect sizes say how much it matters. Cohen's d, Pearson's r, the odds ratio, and η² are different lenses on the same idea, each tuned to a different design.

How to read d, r, OR, η². Cohen's d is the gap between two group means measured in pooled SDs - d = 0.2 small, 0.5 medium, 0.8 large. Pearson's r measures linear association from −1 to +1; r = 0.10 / 0.30 / 0.50 are Cohen's small/medium/large. Odds ratio compares the odds of an outcome between groups; OR = 1 means no effect. η² is the proportion of total variance explained by a factor (ANOVA); η² = 0.01 / 0.06 / 0.14 are small/medium/large.

Picking the right one. Two groups, continuous outcome → d (or g for small samples). Continuous × continuous → r. Two groups, binary outcome → OR (or NNT for clinicians). Multi-group ANOVA → η² or Cohen's f. Want a probability you can read out loud? CLES: P(a random X1 beats a random X2).

Why convert. Meta-analysis pools studies that report different metrics; reviewers ask “what's the d here?” for a study reported as r; an OR is more publishable in clinical journals while d is the field standard in psych. Each conversion has assumptions - especially the d↔OR bridge, which assumes a logistic latent variable.

8 effect sizes · one tool · d · g · r · OR · η² · Cohen's f · NNT · CLES · Runs in your browser

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📝 t-test → d

Two-group continuous outcome. Convert d to r and CLES so you can read the result several ways.

Output
R code RUNNABLE
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Effect-size visual INTERACTIVE

        

          
          

          

          

        

      

    

  


  

    
Inference

    

  


  

    
      Read more
      The conversion formulas
    
    


      

        
r = d / √(d² + 4)        (equal n)
r = d / √(d² + a)        (a = (n₁+n₂)²/(n₁n₂))
d = 2r / √(1 − r²)

        
d ↔ r. Two ways of saying the same thing about a two-group contrast: as a standardized mean difference or as a point-biserial correlation between group membership and the outcome. The 4 in d→r is the equal-n simplification of (n₁+n₂)²/(n₁n₂). For very unbalanced designs use the second formula.

      


      

        
d = ln(OR) · √3 / π
OR = exp(d · π / √3)

        
d ↔ OR. Hasselblad & Hedges (1995). Assumes a logistic latent variable underlying the binary outcome. Multiply ln(OR) by √3/π ≈ 0.5513 to land on Cohen's d.

      


      

        
g = d · J,  J = 1 − 3/(4N − 9)
N = n₁ + n₂

        
d ↔ Hedges' g. Hedges' bias correction. With small N, Cohen's d slightly overestimates the population effect; J shaves it down. By N ≈ 50 the correction is <1%, so g is mainly a small-sample fix.

      


      

        
f = √(η² / (1 − η²))
η² = f² / (1 + f²)

        
η² ↔ Cohen's f. Cohen's f is the SD of group means divided by the within-group SD - a generalisation of d to k groups. η² is the proportion of variance explained. Cohen's f = 0.10 / 0.25 / 0.40 are small/medium/large; the matching η² thresholds are 0.01 / 0.06 / 0.14.

      


      

        
CLES = Φ(d / √2)             (probability of superiority)
NNT  ≈ 1 / Φ(d)                (small-effect approximation)

        
OR ↔ NNT · d ↔ CLES. NNT is the number you need to treat to gain one extra successful outcome - usually computed from absolute risk reduction (1 − RR) · p₁. CLES (the “common language effect size”) is the probability that a random observation from group 1 exceeds a random observation from group 2; for a Cohen's d, that's Φ(d/√2).

      


    

  


  

    
      Caveats
      When this is the wrong tool
    
    

      

        
If you have…

        
Use instead

        
Skewed / heavy-tailed continuous outcome

        
Cohen's d assumes near-normal data with comparable SDs. For non-normal outcomes prefer the rank-biserial correlation, Cliff's delta, or the U-based CLES from the Mann–Whitney test.

        
Clustered data (students in classes, patients in clinics)

        
The pooled-SD d ignores the cluster structure and underestimates uncertainty. Use a multilevel d that conditions on the cluster (Hedges 2007), or report the level you care about explicitly.

        
Categorical predictor with >2 levels

        
Don't pretend a multi-category contrast is a single d. Report η² (or ω²) for the omnibus effect, and Cramér's V or pairwise d for follow-ups.

        
Repeated measurements on the same units

        
The naive d uses an SD that mixes between- and within-subject variation; switch to d_z (standardized by the difference SD) or report change-score d.

        
Very rare outcomes (p < 0.01) with OR/RR/NNT

        
The d ↔ OR formula assumes a logistic latent variable and stable variance; with rare outcomes the conversion gets brittle. Report OR and absolute risk side by side.

        
Time-to-event / survival data

        
Use hazard ratios from a Cox model. d, r, and OR all wash out the time dimension.

      

    

  


  

    
Further reading

    
    
Conversions: d↔r exact under equal n; d↔OR via Hasselblad–Hedges (logistic latent); g uses Hedges' J; CIs for d use Hedges–Olkin SE; CIs for r use Fisher z.