New simple bounds for standard normal distribution function

• Published in: Communications in statistics. Simulation and computation Vol. 54; no. 7; pp. 2762 - 2769
• Main Authors: Ananbeh, Enas A., Eidous, Omar M.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 03.07.2025
• Subjects: Cumulative distribution function; Error function; Maximum absolute error; Mean absolute error; Normal distribution; Q-function; Upper and lower bounds
• ISSN: 0361-0918; 1532-4141
• DOI: 10.1080/03610918.2024.2326596

This 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.