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The cumulative normal distribution

The cumulative normal distribution for A logistic approximation, This paper develops a logistic approximation to the cumulative normal distribution. Although the literature contains a vast collection of approximate functions for the normal distribution, they are very complicated, not very accurate, or valid for only a limited range. This paper proposes an enhanced approximate function. When comparing the proposed function to other approximations studied in the literature, it can be observed that the proposed logistic approximation has a simpler functional form and that it gives higher accuracy, with the maximum error of less than 0.00014 for the entire range. This is, to the best of the authors’ knowledge, the lowest level of error reported in the literature. The proposed logistic approximate function may be appealing to researchers, practitioners and educators given its functional simplicity and mathematical accuracy.

The most important continuous probability distribution used in engineering and science is perhaps the normal distribution. The normal distribution reasonably describes many phenomena that occur in nature. In addition, errors in measurements are extremely well approximated with the normal distribution. In 1733, DeMoivre developed the mathematical equation of the normal curve it provided a basis on which much of the theory of inductive statistics is founded. The normal distribution is often referred to as the Gaussian distribution, in honor of Karl Friedrich Gauss, who also derived its equation from a study of error in repeated measurements of an unknown quantity. The normal distribution finds numerous applications as a limiting distribution. Under certain conditions, the normal distribution provides a good approximation to binomial and hypergeometric distributions. In addition, it appears that the limiting distribution of sample averages is normal. This provides a broad base for statistical inference that proves very valuable in estimation and hypothesis testing. If a random variable X is normally distributed with mean μ and variance σ2, its probability density function is defined as

Unfortunately, there is no closed-form solution available for the above integral, and the values are usually found from the tables of the cumulative normal distribution. From a practical point of view, however, the standard normal distribution table only provides the cumulative probabilities associated with certain discrete z-values. When the z-value of interest is not available from the table, which frequently happens, practitioners often guess its probability by means of a linear interpolation using two adjacent z-values, or rely on statistical software.

In order to rectify this practical inconvenience, a number of approximate functions for a cumulative normal distribution have been reported in the research community. The literature review indicates, however, that they are mathematically complicated, do not have much accuracy, and lack validity when the entire range of z-values is considered. In order to address these shortcomings, this paper develops a logistic approximate function for the cumulative normal distribution. The mathematical form of the proposed function is much simpler than the majority of other approximate functions studied in the literature. In fact, probabilities can be even obtained by using a calculator. Further, the accuracy of the proposed function is higher than with the other approximate functions.

The remainder of the paper is organized as follows. In section 2, the existing literature on approximations to a cumulative normal distribution is discussed. Section 3 first discusses a logistic distribution and notes the similarities and differences between the logistic and normal distributions are noted. Section 3 then proposes the modified logistic approximate function by numerically identifying polynomial regression coefficients in such a way that the absolute maximum deviation between the cumulative distribution and the modified logistic function is minimized. Section 4 evaluates the accuracy of the proposed approximate function, and section 5 discusses and concludes about the results of this paper.

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