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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
Step-by-step solution
🎯 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
$$d = \frac{M_1-M_2}{SD_{pooled}},\quad SD_{pooled}=\sqrt{\frac{(n_1-1)SD_1^2+(n_2-1)SD_2^2}{n_1+n_2-2}}$$
Worked example — Drug vs. Control (n=15 each, M₁=82, M₂=76, SD₁=6.5, SD₂=5.9)
$$SD_{pooled}=\sqrt{\frac{14(6.5)^2+14(5.9)^2}{28}}=\sqrt{\frac{592+487}{28}}=\sqrt{38.5}=6.21$$
$$d=\frac{82-76}{6.21}=\frac{6}{6.21}=\mathbf{0.97}\quad(\text{large})$$
② Eta-squared η² — for ANOVA (proportion of variance explained)
$$\eta^2=\frac{SS_{between}}{SS_{total}},\quad \omega^2=\frac{SS_{between}-df_{between}\cdot MS_{within}}{SS_{total}+MS_{within}}\;(\text{less biased})$$
$$\text{Example: }SS_{between}=180,\;SS_{total}=520 \;\Rightarrow\; \eta^2=\frac{180}{520}=\mathbf{.346}\quad(\text{large})$$
③ Cramér's V — for chi-square (association between categorical variables)
$$V=\sqrt{\frac{\chi^2}{n\cdot\min(r-1,\;c-1)}}$$
$$\text{Example: }\chi^2=8.64,\;n=45,\;2\times2\text{ table} \;\Rightarrow\; V=\sqrt{\frac{8.64}{45\times1}}=\sqrt{.192}=\mathbf{.438}\quad(\text{large})$$
④ r — for Mann-Whitney / Wilcoxon (non-parametric)
$$r=\frac{|z|}{\sqrt{N}}$$
$$\text{Example: }z=-2.45,\;N=30 \;\Rightarrow\; r=\frac{2.45}{\sqrt{30}}=\frac{2.45}{5.48}=\mathbf{.447}\quad(\text{medium-large})$$
| 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 |
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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.