Moving Average Control Charts with Span 2 and Klein’s 2-of-2 Run Rule

The X-bar control chart remains the most widely used scheme in industry due to its simplicity, despite its limited performance for detecting small process mean shifts. Klein’s 2-of-2 supplementary run rule improves this performance while preserving the familiar Shewhart structure. Similarly, the moving average control chart offers a practical balance between simplicity and detection capability, with MA(2) being one of its simplest forms. Motivated by the industrial preference for simple and effective monitoring schemes, this paper proposes the MA(2)-Klein control chart, combining the MA(2) statistic with Klein’s 2-of-2 rule. Control limits are determined through numerical optimization under fixed ARL0. Performance is evaluated using a second-order Markov chain approximation, avoiding simulation and augmented state-space discretization. Results show strong agreement with Monte Carlo estimates and indicate that the proposed chart outperforms both the standard MA(2) and the X-bar-Klein chart for small mean shifts while preserving practical implementation.