Shewhart Control Charts with Guaranteed In-Control Performance

The in-control performance of the Shewhart X ̅ and S2 control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In our paper we present simple formulas for the required control limits so that practitioners do not have to use the bootstrap method. An assumption of normality is required. The advantage of our proposed method is in its simplicity; there is no bootstrapping and the control chart constants do not depend on the Phase I sample data.