The exact run length distribution and design of the Shewhart X Chart with estimated parameters based on median run length

This paper investigates the Shewhart (Formula presented.) chart with estimated parameters. When parameters are unknown, we show that the average run length is a confusing performance measure and that the median run length, which is a better representation of the central tendency associated with different shapes of the run length distribution, is more intuitive. A new design model for the Shewhart (Formula presented.) chart with estimated parameters is developed. With this proposed design model, the Shewhart (Formula presented.) chart for both known and estimated parameters, even with a small number of Phase-I samples, will have an almost similar sensitivity for a specified shift