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?

  • 2
    $\begingroup$ 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.) $\endgroup$
    – Nick Cox
    Mar 13, 2016 at 6:52

1 Answer 1


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.

  • 1
    $\begingroup$ Ah..thought it must be something "pragmatic"..thank you very much!! $\endgroup$
    – Pugl
    Sep 7, 2015 at 10:33

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