⚠️ Example not found

This example does not exist or may have been moved.

← Browse all examples

🎯

Effect Size

Effect Sizes — Cohen's d, η², Cramér's V, and r

Four effect size measures: Cohen's d=0.97 (large), η²=.346, Cramér's V=.438, r=.447. Reference table for thresholds.

📊 d=0.97 (large), η²=.35, V=.44

← All Examples 📄 Full Examples Page ▶ Open in MindStat App

🎯 16 · Effect Sizes — Cohen's d, η², Cramér's V, r

Calculating & Interpreting Four Common Effect Size Measures

Effect Size

Effect size quantifies the practical magnitude of a result, independent of sample size. Always report effect size alongside p-values in APA-7 manuscripts.

① Cohen's d — for t-tests (difference between two means)

Pooled-SD Formula

d = \frac{M_{1} - M_{2}}{S D_{p o o l e d}}, S D_{p o o l e d} = \sqrt{\frac{(n_{1} - 1) S D_{1}^{2} + (n_{2} - 1) S D_{2}^{2}}{n_{1} + n_{2} - 2}}

Worked example — Drug vs. Control (n=15 each, M₁=82, M₂=76, SD₁=6.5, SD₂=5.9)

S D_{p o o l e d} = \sqrt{\frac{14 (6.5)^{2} + 14 (5.9)^{2}}{28}} = \sqrt{\frac{592 + 487}{28}} = \sqrt{38.5} = 6.21

d = \frac{82 - 76}{6.21} = \frac{6}{6.21} = 0.97 (large)

② Eta-squared η² — for ANOVA (proportion of variance explained)

η^{2} = \frac{S S_{b e t w e e n}}{S S_{t o t a l}}, ω^{2} = \frac{S S_{b e t w e e n} - d f_{b e t w e e n} \cdot M S_{w i t h i n}}{S S_{t o t a l} + M S_{w i t h i n}} (less biased)

Example: S S_{b e t w e e n} = 180, S S_{t o t a l} = 520 \Rightarrow η^{2} = \frac{180}{520} = .346 (large)

③ Cramér's V — for chi-square (association between categorical variables)

V = \sqrt{\frac{χ^{2}}{n \cdot min (r - 1, c - 1)}}

Example: χ^{2} = 8.64, n = 45, 2 \times 2 table \Rightarrow V = \sqrt{\frac{8.64}{45 \times 1}} = \sqrt{.192} = .438 (large)

④ r — for Mann-Whitney / Wilcoxon (non-parametric)

r = \frac{| z |}{\sqrt{N}}

Example: z = - 2.45, N = 30 \Rightarrow r = \frac{2.45}{\sqrt{30}} = \frac{2.45}{5.48} = .447 (medium-large)

Measure	Use with	Small	Medium	Large
Cohen's d	t-tests	0.20	0.50	0.80
η² (eta-squared)	ANOVA	.01	.06	.14
η²p (partial)	Factorial ANOVA	.01	.06	.14
Cramér's V	Chi-square	.10	.30	.50
r (Pearson/Spearman)	Correlation	.10	.30	.50
r (z/√N)	Mann-Whitney	.10	.30	.50

💡 Cohen's (1988) thresholds are benchmarks, not rules. A 'small' effect can be critically important in medicine; a 'large' effect may be trivial in engineering. Always interpret effect sizes in context.

⚠️ Example not found

Effect Sizes — Cohen's d, η², Cramér's V, and r

🎯 16 · Effect Sizes — Cohen's d, η², Cramér's V, r

Related Calculators

Related Examples