An entropic framework for the normal distribution in capability analysis

Abstract

The intent of setting a threshold value on C_p,C_pk or C_pm for a quality characteristic in a manufacturing process is to constrain the percentage oi nonconforming parts. One of the important assumptions to the implementation of such a threshold value is that the quality characteristic is normally distributed. The purpose of this article is to demonstrate, using an entropic framework, that given a threshold constraint on the numerical value of the population capability index, C_pm, the most general distribution for our process is, in fact, the normal distribution. Furthermore, this most general distribution, based solely on the principle of maximum entropy over a constrained region, can be found prior to data collected on the process. In addition, if the derived normal distribution is treated as a prior distribution, then using the Kullback-Leibler (direct) divergence criteria and sample mean and variance values, a posterior or updated distribution can be found. It turns out that this updated distribution is again normal. The maximum entropy construct provides a mathematical foundation foi the traditional normality assumption underlying capability analysis. Furthermore, it is suggested that the minimum divergence principle could be used to assess distributional changes in underlying process behavior.

Keywords:

• capability index
• maximum entropy
• minimum divergence: normal distubution