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Cohen's d (Effect Size)

Formula

\[ d = \frac{\bar x_1-\bar x_2}{s_p} \]
\[ s_p = \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}} \]

Parameters

  • \(\bar x_1,\bar x_2\): group means
  • \(s_p\): pooled standard deviation

What it means

Cohen's d standardizes the difference between two means to express practical effect size.

What it's used for

  • Comparing effect magnitude across studies or metrics.
  • Complementing p-values in experiments.

Key properties

  • Scale-free interpretation of mean difference.
  • Useful for power analysis and sample size planning.

Common gotchas

  • Heuristic labels (small/medium/large) are context-dependent.
  • Use paired or unequal-variance variants when appropriate.

Example

A mean uplift of 2 points with pooled SD 10 gives \(d=0.2\), a small standardized effect.

How to Compute (Pseudocode)

Input: two sample groups (or summary stats)
Output: Cohen's d effect size

compute group means and standard deviations
compute pooled standard deviation (or the variant-specific denominator)
return standardized mean difference d

Complexity

  • Time: Typically \(O(n)\) to compute summary statistics from raw data (or \(O(1)\) from provided summaries)
  • Space: \(O(1)\) extra space for streaming summary computations
  • Assumptions: Exact formula depends on independent/paired design and pooled vs unpooled standardization choice

See also