Table 1.

Formulas for the calculation of commonly used effect sizes and corresponding sampling variance

Effect size measure	Study design	Statistics needed	Effect size formula	Sampling variance
Standardized mean difference	Independent groups design	${\overline{X}}_1$, ${\overline{X}}_2,$ $S_1^2,S_2^2,$ $n_1,n_2$	$g_{\mathit{ig}}=\frac{{\overline{X}}_1-{\overline{X}}_2}{S_{\mathit{pooled}}}\bullet \left[1,-,\frac{3}{4\bullet \left(n_1,+,n_2\right)-9}\right]$ where $S_{\mathit{pooled}}=\sqrt{\frac{\left(n_1,-,1\right)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}}$	$S_{g_{\mathit{ig}}}^2=\left[\frac{n_1+n_2}{n_1\bullet n_2},+,\frac{g^2}{2(n_1+n_2)}\right]\bullet {\left[1,-,\frac{3}{4\bullet \left(n_1,+,n_2\right)-9}\right]}^2$
	Matched group design (1)	${\overline{X}}_{\mathit{pre}},{\overline{X}}_{\mathit{post}},S_{\mathit{difference}}$, $r_{pre-post}$	$g_{r{m}_{(1)}}=\frac{{\overline{X}}_{\mathit{post}}-{\overline{X}}_{\mathit{pre}}}{S_{\mathit{within}}}\bullet \left[1,-,\frac{3}{4\bullet \left(n,-,1\right)-1}\right]$ where $S_{\mathit{within}}=\frac{S_{\mathit{difference}}}{\sqrt{2(1-r)}}$	$S_{g_{r{m}_{(1)}}}^2=\left(\frac{1}{n},+,\frac{{g^2}_{r{m}_{\left(1\right)}}}{2n}\right)\left(2,\left(1,-,r\right)\right)\bullet {\left[1,-,\frac{3}{4\bullet \left(n,-,1\right)-1}\right]}^2$
	Matched group design (2)	${\overline{X}}_{\mathit{pre}},{\overline{X}}_{\mathit{post}},S_{\mathit{pre}}$,$r_{pre-post}$	$g_{r{m}_{(2)}}=\frac{{\overline{X}}_{\mathit{post}}-{\overline{X}}_{\mathit{pre}}}{S_{\mathit{pre}}}\bullet \left[1,-,\frac{3}{4\bullet \left(n,-,1\right)-1}\right]$	$S_{g_{{\mathit{rm}}_{(2)}}}^2=\left[\frac{2(1-r)}{n}\right]\left(\frac{n-1}{n-3}\right)\left[1,+,\frac{n{g}_{{\mathit{rm}}_{(2)}}^2}{2(1-r)}\right]{\left[1,-,\frac{3}{4\bullet \left(n,-,1\right)-1}\right]}^2-g_{{\mathit{rm}}_{(2)}}^2$
	Independent groups with pre- and post-test measures (1)	${\overline{X}}_{1,pre}$, ${\overline{X}}_{2,pre},{\overline{X}}_{1,post}$, ${\overline{X}}_{2,post},S_{1,difference,}$, $S_{2,difference,}$,$r_{1,pre-post}$,  $r_{2,pre-post}$	$g_{{\mathit{igpp}}_{(1)}}=g_{{\mathit{rm}}_{(1)}}^E-g_{{\mathit{rm}}_{(1)}}^C$	$S_{g_{{\mathit{igpp}}_{(1)}}}^2=S_{g_{{\mathit{rm}}_{(1)}}^E}^2+S_{g_{{\mathit{rm}}_{(1)}}^C}^2$
Standardized mean difference (cont.)	Independent groups with pre- and post-test measures (2)	${\overline{X}}_{1,pre}$, ${\overline{X}}_{2,pre},$ ${\overline{X}}_{1,post}$, ${\overline{X}}_{2,post},S_{1,pre,}$, $S_{2,pre,}$, $r_{1,pre-post}$,  $r_{2,pre-post}$	$g_{{\mathit{igpp}}_{(2)}}=g_{{\mathit{rm}}_{(2)}}^E-g_{{\mathit{rm}}_{(2)}}^C$	$S_{g_{{\mathit{igpp}}_{(2)}}}^2=S_{g_{{\mathit{rm}}_{(2)}}^E}^2+S_{g_{{\mathit{rm}}_{(2)}}^C}^2$
Risk ratio	Prospective/Longitudinal studies	$f_1,f_0,$ $n_1,n_0$ (see 2×2 table below)	$RR=\frac{p_1}{p_0}$ where $p_1=\frac{f_1}{n_1}$and$p_0=\frac{f_0}{n_0}$	${S^2}_{Ln(RP)}=\frac{1-p_1}{p_1\bullet n_1}+\frac{1-p_0}{p_0\bullet n_0}$
Odds ratio	Cross-sectional	$f_1,f_0,f_2,f_3$ (see 2×2 table below)	$OR=\frac{f_1\bullet f_2}{f_0\bullet f_3}$	${S^2}_{\text{log}\left(O,R\right)}=\frac{1}{f_0}+\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}$
Psychometric effect sizes	McDonald’s ω	${\lambda }_i$,${\epsilon }_i$	$\omega =\frac{{\left({\sum }_{i=1}^I,{\lambda }_i\right)}^2}{{\left({\sum }_{i=1}^I,{\lambda }_i\right)}^2+{\sum }_{i=1}^I{\epsilon }_i}$	No consensus (use e.g., bootstrap)
	Validity correction for unreliability	$r_{\mathit{xy}}$,$r_{xx\prime }$,$r_{yy\prime }$	${\rho }_{\mathit{xy}}=\frac{r_{\mathit{xy}}}{\sqrt{r_{xx\prime }·r_{yy\prime }}}$	No consensus (use e.g., bootstrap)
	Validity correction for range restriction	$r$,$r^u$	${\rho }^u=\frac{r}{a}$ where $a=\frac{r}{r^u}$	$S_{{\rho }^u}^2=\frac{S_r^2}{a^2}$

${\overline{X}}_1$= mean of the first group;${\overline{X}}_2$= mean of the second group;$S_1^2$= variance of the first group;$S_2^2$= variance of the second group;

$n_1$= sample size of the first group;$n_2$= sample size of the second group;${\overline{X}}_{\mathit{pre}}$= mean of the pre-test;${\overline{X}}_{\mathit{post}}$= mean of the post-test;$S_{\mathit{difference}}$= standard deviation of the difference scores (i.e., difference between pre-test and post-test scores);$S_{\mathit{pre}}$= standard deviation of the pre-test scores;$r_{pre-post}$= correlation between pre-test and post-test scores;${\overline{X}}_{1,pre}$= mean score of first group at pre-test;${\overline{X}}_{2,pre}$= mean score of second group at pre-test;${\overline{X}}_{1,post}$= mean score of first group at post-test;${\overline{X}}_{2,post}$= mean score of second group at post-test;$S_{1,difference}$= standard deviation of the difference scores in the first group;$S_{2,difference}$= standard deviation of the difference scores in the second group;$r_{1,pre-post}$= correlation between pre-test and post-test scores in the first group;$r_{2,pre-post}$= correlation between pre-test and post-test scores in the second group;${\lambda }_i$= standardized factor loading for item i in 1 to I;${\epsilon }_i$= error variance for item i in 1 to I;$r_{\mathit{xy}}$= correlation between scale scores and criterion;$r_{xx\prime }$= reliability of scale scores;$r_{yy\prime }$= reliability of the criterion;$r$: observed validity coefficient;$r^u$= unrestricted validity coefficient;$S_r^2$= variance of the observed validity coefficient