Does anyone know what the Inverse Cumulative Distribution Function of Normal Distribution is? Does it have a closed form expression? I did not find any good answer using Google.


1 Answer 1


There's no closed form expression for the inverse cdf of a normal (a.k.a. the quantile function of a normal). It looks like this:

graph showing inverse cdf of standard normal

There are various ways to express the function (e.g. as an infinite series or as a continued fraction), and numerous approximations (which is how computers are able to "calculate" it).

Reasonably accurate approximations are tedious to write and not especially enlightening (except in so far as the general forms convey a little insight into common ways of approximating functions you can't easily obtain in closed form).

If you regard $\text{erf}^{-1}$ or $\text{erf}$ itself as a special function, then it could be written in terms of one of those, but one could as well call $\Phi^{-1}$ a special function and be done in one step.


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