# 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?

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 $$\rho$$.
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.