Cohen’s d Calculator

Cohen's d Calculator

Study Design

Sample 1 (e.g. Treatment / Condition A)

Mean (x̄₁)

Std Dev (s₁)

Sample Size (n₁)

Sample 2 (e.g. Control / Condition B)

Mean (x̄₂)

Std Dev (s₂)

Sample Size (n₂)

Significance Level (α) Used for 95% Confidence Interval ($1 - \alpha = 0.95$)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Standardized Mean Difference

What Is Cohen's d?

Cohen's d is the standardized effect size used to quantify the magnitude of difference between two means in terms of standard deviation units.

While $p$-values evaluate whether an observed difference is likely due to sampling chance, Cohen's $d$ measures practical significance—how large and meaningful the intervention effect actually is, independent of sample size.

When to Use It

Post-t-Test Reporting: Providing effect sizes alongside Independent T-Tests or Paired T-Tests.
Meta-Analyses: Standardizing disparate measurement scales into a common metric across multiple published studies.
Power & Sample Size Planning: Establishing expected effect sizes for prospective power analysis.
Clinical & Practical Relevance: Assessing real-world diagnostic or educational improvement magnitudes.

The Standardized Effect Size Formulas

$$\text{Independent: } d = \frac{\bar{x}_1 - \bar{x}_2}{s_{\text{pooled}}}, \quad s_{\text{pooled}} = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$ $$\text{Hedges' } g = d \times \left(1 - \frac{3}{4(n_1 + n_2) - 9}\right), \quad \text{Glass's } \Delta = \frac{\bar{x}_1 - \bar{x}_2}{s_{\text{control}}}$$ $$SE(d) = \sqrt{\frac{n_1 + n_2}{n_1 n_2} + \frac{d^2}{2(n_1 + n_2)}}, \quad 95\% \text{ CI} = d \pm z_{\alpha/2} \cdot SE(d)$$

Where $s_{\text{pooled}}$ is the pooled sample standard deviation, and Hedges' $g$ applies an exact correction factor for small-sample upward bias ($N < 20$).

Worked Example

Scenario: Group 1 ($n_1=20, \bar{x}_1=25, s_1=5$) vs Group 2 ($n_2=20, \bar{x}_2=20, s_2=5$) at $\alpha=0.05$.

Step 1: Calculate Pooled Standard Deviation ($s_{\text{pooled}}$):
$s_{\text{pooled}} = \sqrt{\frac{19(25) + 19(25)}{38}} = \sqrt{25} = 5.0000$.

Step 2: Calculate Cohen's d:
$d = \frac{25 - 20}{5.0000} = \frac{5}{5} = 1.0000 \text{ (Large Effect)}$.

Step 3: Small-Sample Hedges' g Correction:
$J = 1 - \frac{3}{4(40) - 9} = 1 - \frac{3}{151} \approx 0.9801 \implies g = 0.9801 \times 1.0 = 0.9801$.

Step 4: Standard Error & 95% Confidence Interval:
$SE(d) = \sqrt{\frac{40}{400} + \frac{1^2}{80}} = \sqrt{0.1125} = 0.3354 \implies 95\% \text{ CI} = [0.3426, 1.6574]$.

Standardized Effect Size Complete

Cohen's d Results

Standardized Effect Size (Cohen's d)

d = 1.0000

Large Effect Size (|d| ≥ 0.80)

Cohen's d 1.0000

Hedges' g (Corrected) 0.9801

Glass's Delta (Δ) 1.0000

Pooled SD (s_p) 5.0000

Cohen's U₃ 84.13%

Distribution Overlap 61.71%

Cohen’s d Calculator