An Approximate Formula for the Cumulative $z$-Distribution

Abstract

A straightforward expansion and integration of the frequency function for Fisher's $z$ produces a formula for the probability that $z$ is not exceeded, of which the successive terms decrease rapidly when $n_1$ and $n_2$ are large. It is given in terms of incomplete normal moment functions (or $\chi^2$ probabilities), and as a polynomial in $zN^{1/2}$, where $N$ is the harmonic mean of $n_1$ and $n_2$. This last form is identical with the inverted Cornish-Fisher expansion, originally deduced by quite different methods.