Effects of Standard Deviation Estimation on the X-Bar Control Chart and Adjustments for a Guaranteed In-Control Performance

• Activity Number: 310
• Type: Contributed
• Date/Time: Tuesday, August 2, 2016 : 8:30 AM to 10:20 AM
• Sponsor: Quality and Productivity Section
• Author(s): Felipe Jardim* and Subhabrata Chakraborti and Eugenio Epprecht
• Companies: Pontífice Universidade Católica Rio de Janeiro and University of Alabama and Pontífice Universidade Católica Rio de Janeiro
• Keywords: X-bar Control Chart Performance ; Conditional and Unconditional Performance ; False Alarm Rate ; Control Limits Adjustments ; Guaranteed In-Control Performance ; Average Run Length

Abstract

The X-Bar Control Chart is commonly used for monitoring the mean of a process. Its performance when the process parameters are estimated has been widely discussed in the literature. Most of these studies have focused on the unconditional in-control (IC) run length distribution. However, recent works showed that in the face of parameter estimation, the knowledge of the conditional IC average run length distribution or the conditional false alarm rate (CFAR) distribution may be more useful. To this end, we study the performance of the X-Bar Control Chart where the mean is specified but the standard deviation is unknown and is estimated from a set of Phase I reference data, by providing a closed form expression for the cumulative distribution function (c.d.f.) of the CFAR. Using these expressions, we construct a one-sided prediction interval for the CFAR for several Phase I (reference) sample sizes and show that the minimum number of reference samples that guarantees a desired typical nominal IC performance is large and infeasible in many practical settings. Following up, we propose corrections of the control limits in order to guarantee a desired IC performance for several numbers and sizes of reference samples.