Effect Size (δ)#
The magnitude of a difference or relationship, independent of sample size.
Important
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What it is#
Effect size \(\delta\) measures the magnitude of a difference or relationship — how big an effect is, not merely whether it exists. Where a p-value answers the yes/no question of detectability, effect size carries the practical importance, and it is the key input to power analysis.
Why it matters#
Statistical and practical significance diverge: with a huge sample a 0.1% difference can be “significant” yet trivial. Effect size restores the real-world magnitude, which is why it anchors sample-size planning, results reporting in medicine and psychology, and A/B testing in business.
The common forms#
Cohen’s d (standardised mean difference): \(d = (\bar{X}_1 - \bar{X}_2)/s_p\) with pooled SD \(s_p\); by rule of thumb 0.2 small, 0.5 medium, 0.8 large.
Noncentrality \(\delta\) (power analysis): \(\delta = (\mu - \mu_0)/(\sigma/\sqrt{n})\), essentially the expected t-statistic under \(H_1\) — larger \(\delta\), higher power.
Association: Pearson \(r\), variance-explained \(R^2\), and ANOVA’s \(\eta^2\).
Proportions (A/B): the raw gap \(\delta = p_1 - p_2\), or the standardised \(h = 2\arcsin\!\sqrt{p_1} - 2\arcsin\!\sqrt{p_2}\).
Example#
A one-sample t-test with \(H_0\) mean 100, sample mean 104, \(\sigma = 10\), \(n = 25\) gives \(\delta = (104 - 100)/(10/\sqrt{25}) = 4/2 = 2.0\) — the mean is 2 standard errors from \(H_0\), a very large effect.
Effect size vs p-value#
The crucial contrast: a p-value says whether an effect exists and depends on sample size (large \(n\) makes tiny effects significant), while effect size says how big it is and is independent of sample size. Report them together — significance for detectability, effect size for meaning.
Theme: Statistical Inference & Power · All terminology
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Mind map — connected ideas
Statistical Significance · P-Value (probability value) · Statistical Power · Sample size · Minimum Detectable Lift (MDL) · Hypothesis Testing
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More in Statistical Inference & Power
A Priori Power Analysis · Chi-square (χ²) Test · Clopper–Pearson Interval · Compromise Power Analysis · Confidence Intervals (CIs) · Hypothesis Testing · Kolmogorov–Smirnov (KS) Test · Minimum Detectable Lift (MDL) · P-Value (probability value) · Post Hoc Power Analysis · Power (1 – β) · Power Analysis · Sample size · Significance Level (α)
See also
Source article Adapted (context, re-expressed) in our own words from: Effect Size (δ) (insightful-data-lab.com).