Unveiling the intricacies of the normal distribution: From theory to applications

Abstract

The normal distribution, also referred to as the Gaussian distribution, holds immense significance across multiple domains, from mathematics and physics to engineering and statistics. Beginning with its probabilistic foundations, the normal distribution is characterized by its bell-shaped curve, symmetric nature around the mean, and parameterization by the mean () and standard deviation (). Originating from the seminal works of Abraham de Moivre and further advanced by Gauss and Laplace, its theoretical underpinnings have significantly influenced statistical theory. Laplaces insights regarding errors and the normal distribution paved the way for its widespread adoption in statistical analysis. The normal distribution serves as a powerful tool in various applications, including estimating frequency distributions, analyzing student performance, establishing medical reference values, and forming the basis for numerous statistical methods. This paper explores the profound implications of the normal distribution, encompassing its definition, attributes, historical development, and practical applications, highlighting the enduring importance and versatility of the normal distribution in both theoretical and practical contexts.