Can someone explain the Fisher transformation and why it is used in layman's terms? I am writing a paper on comparing different business cycles and have calculated bilateral correlation coefficients between individual countries and the overall aggregate.  I have read in some of the academic literature that the Fisher transformation is necessary in order to enable a comparison of the means.  If someone could explain, if I were to change my cc's with the Fisher transformation, what I would be doing to them and if it would help with comparison? 
 A: The Fisher transformation https://en.wikipedia.org/wiki/Fisher_transformation  of an estimated correlation coefficient $r$ is
$$
   z= \frac12 \ln\left(\frac{1+r}{1-r}\right).
$$
It is an approximate variance-stabilizing transform, so that its variance which is about $\frac{1}{N-3}$, where $N$ is the sample size,  does not depend on the true underlying value of the correlation coefficient.  This can be used to construct a confidence interval for the correlation coefficient $\rho$.
A modern alternative would be to use the bootstrap.  One of the advantages of the bootstrap, according to Efron, is that it can "find" a variance-stabilizing transform like that above "automatically".
To construct the confidence interval, use the approximation, for sufficiently large $N$, that
$$
   Z  \stackrel{\text{a}}{\sim} \text{N}\left(\frac12\ln\left(\frac{1+\rho}{1-\rho}\right),\frac{1}{N-3}\right), 
$$
to find a confidence interval (on the $z$ scale) of the form $(z-q\frac1{\sqrt{N-3}},z+q\frac1{\sqrt{N-3}})$ where $q$ is the appropriate normal quantile, and invert it by using the inverse function $g$ of the Fisher transform,
$$ g(z)=\frac{e^{2z}-1}{e^{2z}+1},
$$
thus obtaining the confidence interval for the correlation coefficient.
