Comparison of Standard and Individual Limits Phase I Shewhart X, R, and S Charts

Phase I control charts are used as aids in bringing a process into and defining the meaning of a process being in a state of statistical in-control. This is done by looking at the data in retrospect to answer the question 'were these data taken from an in-control process?' Typically the data is collected as independent random samples taken periodically from the output of the process. Commonly it is recommended that each sample be compared via a statistic(s) to a control limit(s) that is (are)function(s) of all the data. We refer to these charts as 'standard limits' charts. An alternative method is to compare each sample with all or some subcollection of the remaining samples. These charts are referred to as 'individual' limits charts. The individual limits charts appear to be attractive in detecting a sample from an out-of-control process when the other samples were taken when the process was in-control. We demonstrate using a simple probability analysis that a standard limits Phase I Shewhart X̄ chart performs better than the individual limits Phase I Shewhart X̄ chart. Furthermore, it is shown that the individual limits Phase I Shewhart R charts can be designed to be equivalent. This also holds for the individual and standard limits Phase I Shewhart S charts.