A piecewise polynomial approximation to the normal CDF

The normal cumulative distribution function

has the unfortunate property of not being expressible in terms of elementary functions; that is, it can’t be computed with functions available on a standard scientific calculator. Some modern calculators, such as the Casio ClassPad and the TI-Nspire (and some of their predecessors) can indeed evaluate this function, but for students without such calculators, they need to look up tables – surely an old-fashioned method!

To overcome this there have been masses of approximations produced over the years; one of the simplest is the logistic approximation:

Note that the integrand is an even function, which means that

if , and so we only have to find an approximation for positive .

However, there is a conceptually even easier function, which dates back to a 1968 paper by John Hoyt:

Here is a plot of the two of them together (created using Maxima and gnuplot):

and here is a plot of their difference:

In fact the absolute difference is 0.01 or less for . This piecewise polynomial function provides a nice quick-and-dirty approximation to the normal CDF.