Practical significance and the modified control chart with estimated parameters

Shewhart charts are still widely used in practice to monitor process quality. When a plotted point falls outside the control limits, an out-of-control (OOC) signal is triggered, the process is halted, and a search for assignable causes begins. However, in practice, even when an OOC signal is issued, the process may still produce an acceptably small proportion of nonconforming items, making it practically capable. In such cases, stopping the process may not be necessary, reducing false alarms and saving resources. The modified X¯ control chart (MCC) was proposed for such capable processes, specifically for monitoring the process mean, assuming known parameters. We consider the realistic case where the standard deviation is unknown and must be estimated. Using an estimated standard deviation adds variation to both the fraction nonconforming (FNC) and the false alarm rate, often inflating them beyond nominal levels. We propose an MCC for monitoring the mean with an estimated standard deviation and derive corrections that yield an adjusted acceptable region and control limits that account for estimation effects. This guarantees the desired in-control performance. We also investigate how these corrections affect the expected FNC and the power to detect process mean shifts. An illustrative example is provided.