An Investigation into the Analysis of Truncated Standard Normal Distributions Using Heuristic Techniques

Standard normal distributions (SND) and truncated standard normal distributions (TSND) have been widely used and accepted methods to characterize the data sets in various engineering disciplines, financial industries, medical fields, management, and other mathematic and scientific applications. For engineering managers, risk managers and quality practitioners, the use of the standard normal distribution and truncated standard normal distribution have particular relevance when bounding data sets, evaluating manufacturing and assembly tolerances, and identifying measures of quality. In particular, truncated standard normal distributions are used in areas such as component assemblies to bound upper and lower process specification limits. This dissertation presents a heuristic approach for the analysis of assembly-level truncated standard normal distributions. This dissertation utilizes unique properties of a characteristic function to analyze truncated assemblies. Billingsley (1995) suggests that an inversion equation aids in converting the characteristic functions for a given truncated standard normal distribution to its corresponding probability density function. The heuristic for the inversion characteristics for a single doubly truncated standard normal distribution uses a known truncated standard normal distribution as a probability density function baseline. Additionally, a heuristic for the analysis of TSND assemblies building from the initial inversion heuristic was developed. Three examples are used to further demonstrate the heuristics developed by this dissertation. Mathematical formulation, along with correlation and regression analysis results, support the alternate hypotheses presented by this dissertation. The correlation and regression analysis provides additional insight into the relationship between the truncated standard normal distributions analyzed. Heuristic procedures and results from this dissertation will also serve as a benchmark for future research. This research contributes to the body of knowledge and provides opportunities for continued research in the area of truncated distribution analysis. The results and proposed heuristics can be applied by engineering managers, quality practitioners, and other decision makers to the area of assembly analysis