How to compute Cohen's d effect size: standardized measure of difference between two groups, formula d = (M_1 - M_2) / SD_pooled, where SD_pooled = sqrt(((n_1-1)*s_1^2 + (n_2-1)*s_2^2) / (n_1+n_2-2)) is pooled standard deviation. Conventional Cohen 1988 thresholds: |d| 0.8 large. For small samples (n < 50) use Hedges' g = d * (1 - 3/(4(n_1+n_2-2)-1)) correcting positive bias. Glass delta = (M_1 - M_2) / SD_2 useful when one group is control. Effect size mandatory in APA, CONSORT, JAMA, NEJM publications alongside p-value.

Effect size formulas

Cohen d: d = (M_1 - M_2) / SD_pooled. Standardizes mean difference in standard deviation units, independent of original scale (age, kg, IQ, etc.). Allows comparisons across studies (meta-analysis). Typical range [-3, +3], beyond values implausible.

Hedges g: g = d * J, J = 1 - 3/(4*df - 1), df = n_1 + n_2 - 2. Hedges-Olkin (1985) exact correction for Cohen d positive bias in small samples. For n_1 = n_2 = 10, J = 0.958 (4.2% underestimate); for total n > 50, J > 0.98 (negligible correction). Preferred by Cochrane Handbook for Systematic Reviews of Interventions.

Glass delta: delta = (M_1 - M_2) / SD_2, where SD_2 is control group standard deviation. Appropriate in designs with one intervention and one control group, particularly when intervention may modify variance (heteroscedasticity). Glass-McGaw-Smith 1981 first educational research meta-analysis.

Complementary indicators: Cohen U_3 = Phi(d) percentage of group 2 below group 1 mean (intuitive). CLES (Common Language Effect Size, McGraw-Wong 1992) = Phi(d/sqrt(2)) probability that random group 1 observation exceeds group 2. NNT (Number Needed to Treat) = 1/(2*CLES - 1) number of subjects to treat for beneficial outcome vs control (Kraemer-Kupfer 2006).

Confidence Interval: SE(d) = sqrt((n_1+n_2)/(n_1*n_2) + d^2/(2(n_1+n_2))), 95% CI = d +- 1.96*SE. For exact intervals (non-central t-distribution) use R MBESS::ci.smd(), Python pingouin.compute_effsize_from_t().

History and context

Jacob Cohen (1923-1998), NYU statistical psychologist, introduced the measure in his foundational book "Statistical Power Analysis for the Behavioral Sciences" (1969, 2nd ed 1988, Lawrence Erlbaum), reacting to excessive emphasis on p-values without practical interpretation. Cohen proposed thresholds 0.2/0.5/0.8 based on examples from social sciences of his time, emphasizing they should be field-relative. Cohen contributed fundamentally to "power analysis" (sample size calculation) and was vocal critic of Null Hypothesis Significance Testing (NHST). Larry Hedges of Northwestern University proposed bias correction g (1981) and founded modern methodological meta-analysis with Olkin (Hedges-Olkin 1985 "Statistical Methods for Meta-Analysis" Academic Press). Gene Glass of Arizona State published first educational research meta-analysis in 1976 on "psychotherapies" and introduced delta. In the 1990s APA Task Force on Statistical Inference (Wilkinson 1999, "Statistical Methods in Psychology Journals") formally recommended effect size as mandatory. CONSORT 2010 (Schulz-Altman-Moher) requires effect size with CI in clinical trials. ICH E9 and FDA EMA regulators confirm. Cochrane Collaboration mandates Hedges g for clinical meta-analyses.

Threshold interpretation and benchmark values table

• |d| < 0.1: negligible, practically undetectable
• 0.1 <= |d| < 0.2: very small (Sawilowsky 2009 extension)
• 0.2 <= |d| < 0.5: small Cohen (e.g., gender on newborn height 0.2)
• 0.5 <= |d| < 0.8: medium (e.g., CBT depression 0.6)
• 0.8 <= |d| < 1.2: large (e.g., SSRIs vs placebo severe 0.8-1.0)
• |d| >= 1.2: very large (Sawilowsky 2009)
• |d| >= 2.0: huge (e.g., adult vs newborn weight 4.0)
• Cohen U_3 50% for d=0, 84% for d=1, 98% for d=2
• CLES 0.5 for d=0, 0.76 for d=1, 0.92 for d=2
• Field-specific: Bosco et al 2015 management/HR median d=0.40

Worked numerical examples

Example 1 - cognitive behavioral therapy depression. APA-validated study: CBT group (n_1=40) Beck Depression Inventory post-intervention M_1 = 12, SD_1 = 6; waitlist (n_2=40) M_2 = 18, SD_2 = 7. SD_pooled = sqrt((39*36 + 39*49)/78) = sqrt(42.5) = 6.52. Cohen d = (12-18)/6.52 = -0.92 large. Hedges g = -0.92 * 0.990 = -0.91. 95% CI [-1.38; -0.46]. Cohen U_3 = 17.9% (only 17.9% control below treatment mean). NNT = 2.8: treat 3 patients for 1 benefit. APA Division 12 evidence-based practice.

Example 2 - Mediterranean diet weight PREDIMED. NEJM 2013 study: MedDiet group (n_1=2543) BMI delta -0.9, SD 2.1; low-fat control (n_2=2450) BMI delta -0.2, SD 2.0. d = (-0.9 - (-0.2))/2.05 = -0.34 small. Significant difference (p<0.001) but modest effect size. 95% CI [-0.40; -0.29]. Moderate effect size explains population results: NNT = 25 persons per year intervention for -1 BMI.

Example 3 - APA cognitive test. WAIS-IV IQ comparison: high-SES (n_1=200) M_1 = 105, SD = 14; low-SES (n_2=200) M_2 = 95, SD = 14. d = (105-95)/14 = 0.71 medium. SD pooled = 14, 95% CI [0.51; 0.91]. Effect size compatible with literature socioeconomic gradient cognition (Hackman-Farah 2009, Lancet Reviews 2018).

Example 4 - educational intervention US schools. NCES 2024: experimental classes Montessori method (n_1=300) math test M_1 = 78, SD = 12; traditional classes (n_2=300) M_2 = 73, SD = 14. d = 5/13.04 = 0.38 small-medium. 95% CI [0.22; 0.54]. Hattie 2009 meta-analysis educational interventions: hinge point d=0.40 for visible learning. NCES updates curriculum.

Example 5 - SSRI psychiatric pharmacology. Meta-analysis Cipriani et al 2018 Lancet (522 RCTs, 116,477 patients): SSRIs vs placebo moderate-severe depression mean d = 0.30, 95% CI [0.26; 0.34]. Small effect size: clinical vs statistical significance discussion. FDA, EMA guidelines: risk-benefit balance for specific severity.

Limits and considerations

1. d sensitive to outliers: use robust effect size (median, MAD-based) for skewed data.
2. Heteroscedasticity: if SD_1 << SD_2 use Glass delta or pooled SD with Welch correction.
3. Cohen thresholds 0.2/0.5/0.8 generic: in some fields (educational) 0.4 large, in others (medical) 0.2 may be clinically relevant.
4. Positive effect size doesn't imply causality: design (RCT vs observational) separate.
5. Hedges g preferred over d for n < 50.
6. 95% CI mandatory APA, CONSORT, COSMIN for precision interpretation.
7. For paired designs use d_z = M_diff/SD_diff (Lakens 2013 considerations).
8. Multivariate effect size: Mahalanobis D, eta-squared for ANOVA, R^2 regression.
9. Meta-analysis: weight studies by inverse variance, I^2 heterogeneity, DerSimonian-Laird random-effects.
10. Cumming 2014 "New Statistics" recommends focus on CI rather than p-values.

Common errors

1. Reporting d without 95% CI (non-compliant APA, JAMA).
2. Confusing Cohen d with t-statistic (correlated but different).
3. Using d instead of g for small samples (uncorrected positive bias).
4. Applying Glass delta without justifying choice of control SD_2.
5. Interpreting d=0.2 as "non-significant" without CI (may be statistically significant).
6. Summing d across studies without formal meta-analysis (heterogeneity ignored).
7. Confusing d with r (Pearson correlation): conversion r = d/sqrt(d^2+4) for equal groups.
8. Computing effect size on groups with drastically different variances.
9. Reporting effect size ignoring clinical/practical importance (MID Minimum Important Difference).
10. Using d for binary outcomes (prefer OR, RR, NNT).

Practical applications

1. Clinical trials: FDA RCTs, ICH E9, CONSORT 2010 effect size mandatory.
2. Meta-analysis: Cochrane Reviews Hedges g, RevMan software.
3. Psychology: APA tests, Likert scales, intervention efficacy.
4. Educational research: NCES, OECD PISA, Hattie visible learning.
5. Sport science: training intervention USOPC, fitness peripherals.
6. Marketing: AB testing conversion, customer satisfaction shift.
7. Pharmacology: dose-response, FDA/EMA approval thresholds.
8. Scientific publication: APA Manual 7ed, JAMA, NEJM, Lancet.
9. HR/Industrial: training programs, SHRM certification.
10. Sociology: Census group differences (gender, region, class).

Support

Questions: info@calcoi.com. Educational tool for effect size. For professional applications consult Cohen "Statistical Power Analysis" 2nd ed (Erlbaum 1988), Hedges-Olkin "Statistical Methods for Meta-Analysis" (Academic 1985), R packages effsize/MBESS/metafor, Python pingouin compute_effsize.

Updated July 2026