Process Control for Non-Normal Populations Based on an Inverse Normalizing Transformation

Abstract

When the process distribution is non-normal, traditional Shewhart charts may not be applicable. A common practice in such cases is to normalize the process data by applying the Box-Cox power transformation. The general effectiveness of this transformation implies that an inverse normalizing transformation (INT) may also be effective, namely: a power transformation of the standard normal quantile may deliver good representation for the original process data without the need to transform them. Using the Box-Cox transformation as a departure point, three INTs have recently been developed and their effectiveness demonstrated. In this paper we adapt one of these transformations to develop new schemes for process control when the underlying process distribution is non-normal. The adopted transformation has three parameters, and these are identified either by matching of the median, the mean and the standard deviation (Procedure I),or by matching of the median, the mean and the mean of the log (of the original monitoring statistic; Procedure II). Given the low mean-squared-errors (MSEs) associated with sample estimates of these parameters, the new transformation may be used to derive control limits for non-normal populations without resorting to estimates of third and fourth moments, known for their notoriously high MSEs. Implementation of the new approach to monitor processes with non-normal distributions is demonstrated.