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AbstractThis paper presents new simple lower and upper bounds for the cumulative normal distribution function, Φ(z). The accuracy and closeness of the proposed bounds to the exact Φ(z) are investigated based on the maximum absolute error and the mean absolute error. It is found that the maximum absolute error of the proposed lower bound is 8.55×10−3 and it is 4.1×10−4 for the upper bound. In addition, based on 5001 values between z=0 and z=5 with step 0.001, we found that the mean absolute error is 3.27×10−3 for the lower bound and it is 1.1×10−4 for the upper bound and these two values decrease with increasing the z value. [ABSTRACT FROM AUTHOR]

Due to the widespread applicability and use of the normal distribution, a need has arisen to approximate its cumulative distribution function (cdf). In this article, five new simple approximations to the standard normal cdf are developed. In order to assess the accuracy of the proposed approximations, both maximum absolute error and mean absolute error were used. The maximum absolute errors of the proposed approximations lie between 0.00095 and 0.00946, which is highly accurate if compared to the existing simple approximations and quite sufficient for many real-life applications. Even though simple approximations may not as accurate as complicated ones, they are, though, fairly good when judged vis-à-vis their simplicity. [ABSTRACT FROM AUTHOR]

This paper deals with a new simple one-term approximation to the cumulative distribution function (cdf) of the standard normal distribution which does not have a closed-form representation. The accuracy of the proposed approximation has been evaluated by using the maximum absolute error (MAE), which was about 0.0016 and this accuracy is sufficient for most practical applications. To compare the proposed approximation with some of the existing one-term approximations, the MAE measure was considered. In addition, the difference between the exact and approximation of the normal cdf was computed for specific values of the normal random variable. [ABSTRACT FROM AUTHOR]

This article proposes nine new approximations for the standard normal cumulative distribution function Φ z. In addition, it collects most of the approximations existing in the literature. The accuracy of the proposed approximations is evaluated by using the maximum absolute error and the mean absolute error. The maximum absolute errors fall between 0.0422613 × 10−6 and 950.55 × 10−6, which indicates high accuracy for some of them. A comparison study between the different existing approximations and the proposed approximations is also accomplished. [ABSTRACT FROM AUTHOR]