Journals Information
Mathematics and Statistics Vol. 13(5), pp. 297 - 301
DOI: 10.13189/ms.2025.130504
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Improvements of Some Normal Distribution Function Approximations
Mohammed Obeidat *, Rema Al-Jamal , Ahmad Hanandeh
Department of Statistics, College of Science, Yarmouk University, Jordan
ABSTRACT
The Gaussian distribution and its related functions are widely used in statistics, engineering, and physics, including hypothesis testing, quality control, and digital communications. As the standard normal cumulative distribution function (CDF) lacks a closed-form, accurate approximations are essential for practical applications. This study presents new, highly accurate improvements to existing approximations of the standard normal distribution function while preserving simplicity. The proposed methodology refines current formulas by introducing additional parameters and systematically optimizing them using R-based iterative procedures guided by the Kolmogorov-Smirnov statistic to minimize maximum and mean absolute errors. Evaluations across a dense grid confirm the precision and robustness of the proposed models. Results demonstrate significant improvements, with one enhancement reducing the maximum error by a factor of 4.86 compared to previous models, while other formulas show notable accuracy gains. The improved models remain straightforward to implement, making them suitable for real-time engineering and applied statistical computations. A practical application is demonstrated by calculating the bit error rate (BER) in Binary Phase Shift Keying (BPSK) systems under Gaussian noise, where the proposed approximations closely replicate the exact BER curve, confirming their effectiveness in telecommunications system analysis. These improvements provide engineers and scientists with precise, efficient tools, supporting modeling, risk assessment, and system reliability in applications requiring accurate Gaussian evaluations.
KEYWORDS
Standard Normal, Distribution Function, Error Function, Maximum Absolute Error, Mean Absolute Error, Approximation
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mohammed Obeidat , Rema Al-Jamal , Ahmad Hanandeh , "Improvements of Some Normal Distribution Function Approximations," Mathematics and Statistics, Vol. 13, No. 5, pp. 297 - 301, 2025. DOI: 10.13189/ms.2025.130504.
(b). APA Format:
Mohammed Obeidat , Rema Al-Jamal , Ahmad Hanandeh (2025). Improvements of Some Normal Distribution Function Approximations. Mathematics and Statistics, 13(5), 297 - 301. DOI: 10.13189/ms.2025.130504.