Let \(\mu_1\) and \(\mu_2\) be population mean of the two groups. The population mean difference is \[ \Delta = \mu_1-\mu_2. \tag{1.1} \]
We shall compute an estimate \(D\) of the parameter and its variance from studies that used two independent groups AND the studies that used paired groups or matched designs.
Let \(\overline{X}_1\) and \(\overline X_2\) be the sample means of the two independent groups. The sample estimate of \(\Delta\) is just the difference in sample mean \[ D = \overline X_1 - \overline X_2. \tag{1.2} \]
Also, suppose \(S_1\) and \(S_2\) be the sample S.D of the two groups, and \(n_1, n_2\) be the sample sizes. If we assume that \(\sigma_1=\sigma_2=\sigma\), and then \[ V_D = \frac{n_1+n_2}{n_1n_2}S^2_{pooled}, \tag{1.3} \] where \[ S_{pooled} = \bigg[\frac{(n_1-1)S^2_1+(n_2-1)S^2_2}{n_1+n_2-2}\bigg]^{1/2} \tag{1.4} \]
As \(\sigma_1 \ne\sigma_2\): \[ V_D = \frac{S^2_1}{n_1}+\frac{S^2_2}{n_2} \tag{1.5} \]