The X¯ Chart With Estimated Control Limits Is Indeed ARL‐Biased

ABSTRACT The Shewhart chart has remained relevant over a century and continues to be widely studied. This article addresses a specific challenge in the unknown parameter case: The chart's failure to detect process shifts faster than triggering false alarms—a condition known as average run length (ARL)‐bias, where true signals are identified with greater delay than false alarms. This property is supposedly confined to charts that use skewed charting statistics, provided with ‐sigma limits or equal‐tails limits. However, we demonstrate that when the known parameters assumption is violated, the chart using ‐sigma limits is also ARL‐biased. This is counterintuitive to the prevailing belief that the chart detects shifts in the process mean more quickly than it generates false alarms, regardless of whether the process parameters are known or estimated. This uncovering can be useful to alert the practitioners with the severity of ARL‐biasedness and its impact on chart's performance and making recommendations to achieve a satisfactory confidence under the uncertainty of parameter estimation. Furthermore, a proposal for unequal‐distance limits is introduced to account for this property. Case study shows the proposed chart detects positive mean shifts faster than the traditional chart when upward shift detection is prioritized.