The average within-group correlation is estimated as:

$$\rho_{xy_{WG}}=\rho_{xy_{Mix}}\sqrt{\left(d_{x}^{2}p(1-p)+1\right)\left(d_{y}^{2}p(1-p)+1\right)}-\sqrt{d_{x}^{2}d_{y}^{2}p^{2}(1-p)^{2}}$$

where \(\rho_{xy_{WG}}\) is the average within-group correlation, \(\rho_{xy_{Mix}}\) is the overall mixture correlation,
\(d_{x}\) is the standardized mean difference between groups on X, \(d_{y}\) is the standardized mean difference between groups on Y, and
*p* is the proportion of cases in one of the two groups.