Accuracy and speed of seven approximations of the normal distribution function

Abstract

In statistics programs, the evaluation of a normal distribution function is usually accomplished by an approximation algorithm, the most popular of which are those based on Hastings (1955) and adapted by Zelen and Severo (1964). This paper compares seven approximations of the normal distribution function with respect to accuracy and speed of execution on a microcomputer. The results can aid in choosing an appropriate procedure for estimating the probability of a normal deviate (z). 17 from Zelen and Severo require the calculation of the probability density as a first step. Cadwell's approximation was modified slightly by substitution of a new correction term for two terms in the original formula. The modification not only simplifies the approximation but also improves its accuracy somewhat for many values of z. Moran's approximations, which have not yet received much attention, were programmed from Moran's Equations 4 and 5. Unlike Hastings-type approximations, Cadwell's and Moran's approximations do not rely on special constants, although they do use pi, and Moran's approximations use the sine function.Each approximation returns the one-tailed probability of the value of z that is entered. A positive z is assumed as input for all the approximations except Cadwell's (1951), but Line 60 of the driver program, which calculates the percentile rank of z, illustrates how the sign of z can be taken into account.Tests of the Approximations. The approximations were tested on a Radio Shack Color Computer. This machine was selected because it has nine-digit numeric precision and because it is a relatively popular, inexpensive computer that is useful in the psychology laboratory

Approximation of the inverse normal distribution function

“…This paper compares Hastings's approximations with seven other compact approximations of the inverse of the normal distribution function. The results complement Brophy's (1983) comparison of approximations of the normal distribution function. Table 1 shows the approximation methods, in BASIC, together with a short driver program to input a value of p, call a selected method, and print the calculated z.…”

“…If greater accuracy is desired than is provided by the Odeh and Evans (1974) approximation, a more complex procedure can be used. An initial Copyright 1985 Psychonomic Society, Inc. , Milton & Hotchkiss, 1969) or by a Taylor series (e.g., Cunningham, 1969;Hill & Davis, 1973) after calculation of the p corresponding to z' by an approximation of the normal distribution function (see Brophy, 1983). Although highly precise results can be obtained by these methods, the routines are longer and slower than those tested in the present study, and their precision is unnecessary for most practical purposes.…”

Algorithms for fast and precise computation of the normal integral

“…Strecok's method was the basis of Moran's (1980) approximation of the normal distribution, which led to very accurate, compact computer routines (Brophy, 1983). Strecok's method computes the error function as…”

The last normal curve programs

“…The quality of modem personal computers and the NORMAL CURVE PROGRAMS 341 probability-finding algorithms published by Brophy (1983), Moran (1980), Wood (1985), and Wood andWood (1984, 1986) have eliminated the usefulness of the techniques of Kelley (1947Kelley ( -1948 and others having limited accuracy. Because most programmers writing in FORTRAN or COBOL are also fluent in BASIC, translations can readily bemade when programs must bewritten for a special computer or in a language other than BASIC, so that programs introducing error in the 3rd or 4th decimal place should disappear from use.…”

“…Remedies for these two weaknesses or probabilityfinding programs began with the work of Brophy (1983), who recognized that the scientist seeking probabilities would be better served by specific routines than by a single master program and that more powerful mathematics was needed. He examined seven techniques for finding normal curve probabilities and discovered that the technique of Moran (1980) provides 6-to 7-decimal-place accuracy when single-precision arithmetic was used and 9-to to-place accuracy when the 9-digit numerical precision of the Radio Shack Color Computer running Extended Color BASIC was employed.…”