Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

During phase I analysis, the mean and variability of the process are estimated, in order to obtain the control limits for the chart. The effect of the estimation errors in the performance of the chart (phase II) has been widely studied in the literature. However, the papers have been mainly orientated to the in-control performance of the chart. In this paper, the in-control and out-of-control performances of the X over bar control chart are considered at the same time. Therefore, the number of samples to be taken in the phase I analysis is calculated, taking into account the possible deterioration of both metrics. In addition, an adjustment in the control limit is proposed, to reduce this number of samples.