One-term Approximation for Normal Distribution Function

Abstract

This paper presents a one-term approximation to the cumulative normal distribution functions. The absolute maximum error of the proposed approximation is 0.0018 less than 0.003 of Polya's approximation. Comparisons between the proposed approximation and the different approximations with one-term that stated in the literature are given.

A Simple Approximation for Normal Distribution Function

“…That is, Φ 1 ( ) ≈ Φ( ), ∀ ≥ 0 with maximum absolute error 0.0031. In order to improve the Polya's approximation, Eidous and Al-Salaman [4] proved that…”

“…Many efforts have been done to find an approximation for ( ), see for example, Johnson et al [1], Polya [2] and Eidous and Al-Hanandeh [3]. Some of this work was done to replace the tables of cumulative standard normal probabilities that are found in most probability and statistics books.…”

“…A Simple Approximation for Normal Distribution Function with maximum absolute error equal 0.0031. Eidous and Al-Salaman [4] gave the following approximation for ( ),…”

“…with maximum absolute error equal 0.0018. Eidous and Al-Hanandeh [3] gave another approximation for ( ), which is given by,…”

“…The first derivation of ℎ( ) with respect to is ℎ′( 3235 − 0.01575 ) −�0.647 2 −0.021 3 � � 1 − −(0.647 2 −0.0213 ) �.…”