Does transformation of r into Fisher z benefit a meta-analysis? Usually $r$ is transformed into Fisher $z$ to test difference between two $r$ values. But, when a meta-analysis is to be performed, why we should take such a step? Does it correct for measurement error or non-sampling error and why should we assume that $r$ is an imperfect estimate of population correlation? 
 A: There is actually quite a bit of a debate in the literature whether one should conduct a meta-analysis with the raw correlation coefficients or with the r-to-z transformed values. However, leaving aside this discussion, there are really two reasons why the transformation is applied:


*

*Many meta-analytic methods assume that the sampling distribution of the observed outcomes is (at least approximately) normal. When $\rho$ (the true correlation) in a particular study is far away from 0 and the sample size is small, then the sampling distribution of the (raw) correlation becomes very skewed and is not at all well approximated by a normal distribution. Fisher's r-to-z transformation happens to be a rather effective normalizing transformation (even though this is not the primary purpose of the transformation -- see below).

*Many meta-analytic methods assume that the sampling variances of the observed outcomes are (at least approximately) known. For example, for the raw correlation coefficient, the sampling variance is approximately equal to:
$$\text{Var}[r] = \frac{(1-\rho^2)^2}{n-1}$$
In order to actually calculate $\text{Var}[r]$, we must do something about that unknown value of $\rho$ in that equation. For example, we could just plug the observed correlation (i.e., $r$) into the equation. This will give us an estimate of the sampling variance, but this happens to be a rather inaccurate estimate (especially in smaller samples). On the other hand, the sampling variance of an r-to-z transformed correlation is approximately equal to:
$$\text{Var}[z] = \frac{1}{n-3}$$
Note that this no longer depends on any unknown quantities. This is in fact the variance-stabilizing property of the r-to-z transformation (which is the actual purpose of the 
transformation).
