# Domain of Fisher's $z$-transformation

We use the Fisher $z$-transformation for our correlation matrices, s.t. we arrive at approximately normally distributed data.

As you might now, the Fisher $z$-transformation is equivalent to the function

$z = 0.5 \log\left(\frac{1+x}{1-x}\right)$ , $x$ being Pearson's correlation coefficient, ranging between -1 and 1.

Now I have been told that Fisher's $z$ has a domain between -4 and 4 - something which does not make sense to me, since when approximating 1 and -1 for the $x$ values, we arrive at much bigger or smaller values respectively.

Have I been misinformed or am I missing out on something?

• Fisher's $z$ is indeterminate for perfect correlations $r$ of $+1$ or $-1$. That should make sense too. Any uncertainty about whether the result is credible has to be answered otherwise. (Notation $x$ for a correlation seems a little perverse here.) Mar 13, 2016 at 6:52

I suspect that what you heard was meant as a "rule of thumb": under the null hypothesis the Fisher's $z$ should follow a standard normal sampling distribution and values less than $-4$ and more than $4$ are then extremely unlikely.