New Tightness Lower and Upper Bounds for the Standard Normal Distribution Function and Related Functions

Abstract

Most researches interested in finding the bounds of the cumulative standard normal distribution Φ(x) are not tight for all positive values of the argument x. This paper mainly proposes new simple lower and upper bounds for Φ(x). Over the whole range of the positive argument x, the maximum absolute difference between the proposed lower bound and Φ(x) is less than 3×〖10〗^(-4), while it is less than 4.8×〖10〗^(-4) between the proposed upper bound and Φ(x). Numerical comparisons have been made between the proposed bounds and some of the other existing bounds, which showed that the proposed bounds are more compact than most alternative bounds found in the literature.

Communicating educational innovation projects in Latin America mediated by the scaling of complex thinking: Contribution of the UNESCO-ICDE Chair in Mexico

“…This finding conclusively supports the inference that the sample follows a normal distribution in the posttest. The absence of exceptional skewness or kurtosis suggests that the data are distributed symmetrically and more concentrated around the mean, strengthening the validity of the results obtained (Eidous, 2022). The validation of normality provides a solid basis for the application of additional statistical methods and reinforces the reliability of the inferences derived from the analysis of the questionnaire used.…”

“…Evaluate the equality of variances in the pre-& post-test. Wang et al (2023) Normal distributionKnow the distribution of the data after applying the post-test Eidous (2022).…”