Misuse and performance of individuals charts in statistical process control for single parameter distributions of unknown stability

Individuals (XmR) control charts are used in statistical process control for monitoring processes when data occur one at a time, infrequently, or no rational sub-grouping is obvious. Although these charts are based on the assumption of normally distributed data, they sometimes are used as somewhat of an omnibus chart that is robust for all types of data. This thesis discusses the incorrect use of these charts on time between (exponential), number between (geometric), Poisson, and binomial data and compares their performance with other approaches via Monte Carlo simulation. In each case, control limits were calculated using four different methods: 3-sigma limits, probability limits, moving range limits, and moving range limits applied to normalized data. The normalized case was only applied to exponential and geometric data, where the data were transformed to normality and the limits then were found using these transformed data. Each of these methods were analyzed for different amounts of startup data and out-of-control data: post limit (in-control (IC) data used to calculate limits), percent (different percentages of OOC data used to calculate the limits), and alternating (different amounts of alternating IC and OOC data used to calculate limits).;The average run lengths (ARLs) for the exponential and geometric cases were lower for the individuals control chart approach than those for the 3-sigma limits, with differences of up to 17.66 in the exponential case and 72.88 in the geometric case using a parameter of p = 0.25 and post limit OOC data. The ARLs for the transformed XmR method were higher than those for the probability limits approach, with differences as high as 2,597.50 for post limit OOC data and 50 startup data for exponential data. In the binomial and Poisson cases, the ARLs for the XmR approach were higher than the 3-sigma limits approach for post limit OOC data, with ARL differences of up to 44.42 for 100 startup data and lambda = 5 for the Poisson distribution.;Individuals charts are based on the assumption that the data being monitored follow a two parameter normal distribution, where the parameters mu and sigma are usually estimated from the overall sample mean and moving range, respectively. However these charts often are applied to non-normal data, and in particular to distributions with only one estimated parameter that define both the mean and variance. When OOC data or small amounts of IC data are used to separately estimate (unnecessarily) the theoretic variance, this tends to overfit the empirical data (typically with a larger variance), resulting in incorrect limits (typically too wide) and corresponding consequences on ARLs.