Computational Statistics

, Volume 30, Issue 4, pp 1245–1278 | Cite as

Side sensitive group runs \(\bar{{X}}\) chart with estimated process parameters

  • H. W. You
  • Michael B. C. Khoo
  • P. Castagliola
  • Yanjing Ou
Original Paper


The Side Sensitive Group Runs (SSGR) \(\bar{{X}}\) chart integrates the \(\bar{{X}}\) chart and an extended version of the conforming run length chart. The SSGR \(\bar{{X}}\) chart was developed to detect changes in the process mean. The SSGR \(\bar{{X}}\) chart was proven to be effective for detecting small and moderate shifts compared with the \(\bar{{X}},\) synthetic \(\bar{{X}}\) and group runs \(\bar{{X}}\) charts, when process parameters are known. However, in reality, process parameters, such as the in-control mean and standard deviation are rarely known. Therefore, these process parameters are estimated from an in-control Phase I dataset. In this article, we investigate the performance of the SSGR \(\bar{{X}}\) chart, based on the average run length criterion, when process parameters are estimated. It is shown that differences in the chart’s performance exist, when process parameters are known and when they are estimated, due to the variability in estimating the process parameters. A study is conducted to find the minimum number of Phase I samples required (based on several sample sizes) so that the SSGR \(\bar{{X}}\) chart with estimated process parameters behaves approximately the same as its known process parameters counterpart. To facilitate process monitoring and to avoid the need to use large number of samples in the Phase I process, this research develops an optimization procedure using the Scicoslab program to find suitable optimal charting parameters of the SSGR \(\bar{{X}}\) chart with estimated process parameters. This program can be requested from the first author.


Estimation of process parameters Side sensitive group runs \(\bar{{X}}\) chart Group runs \(\bar{{X}}\) chart Synthetic \(\bar{{X}}\) chart \(\bar{{X}}\) chart Conforming run length 



This research is supported by the Universiti Sains Malaysia (USM), Fundamental Research Grant Scheme (FRGS), no. 203/PMATHS/6711322. The authors wish to express their sincere appreciation to the Editor and Referee, for providing useful comments and suggestions, to improve the style and content of this paper.


  1. Capizzi G, Masarotto G (2010) Combined Shewhart-EWMA control charts with estimated parameters. J Stat Comput Simul 80(7):793–807MathSciNetCrossRefMATHGoogle Scholar
  2. Castagliola P, Celano G, Chen G (2009) The exact run length distribution and design of the \(S^{2}\) chart when the in-control variance is estimated. Int J Reliab Qual Saf Eng 16(1):23–38CrossRefGoogle Scholar
  3. Castagliola P, Maravelakis PE (2011) A CUSUM control chart for monitoring the variance when parameters are estimated. J Stat Plan Inference 141(4):1463–1478MathSciNetCrossRefMATHGoogle Scholar
  4. Castagliola P, Zhang Y, Costa AFB, Maravelakis PE (2012) The variable sample size \(\bar{{X}}\) chart with estimated parameters. Qual Reliab Eng Int 28(7):687–699CrossRefGoogle Scholar
  5. Chakraborti S (2007) Run length distribution and percentiles: the Shewhart \(\bar{{X}}\) chart with unknown parameters. Qual Eng 19(2):119–127MathSciNetCrossRefGoogle Scholar
  6. Chakraborti S, Human SW, Graham MA (2008) Phase I statistical process control charts: an overview and some results. Qual Eng 21(1):52–62CrossRefGoogle Scholar
  7. Gadre MP, Rattihalli RN (2004) A group runs control chart for detecting shifts in the process mean. Econ Qual Control 19(1):29–43MathSciNetCrossRefMATHGoogle Scholar
  8. Gadre MP, Rattihalli RN (2007) A side sensitive group runs control chart for detecting shifts in the process mean. Stat Methods Appl 16(1):27–37MathSciNetCrossRefMATHGoogle Scholar
  9. Jensen WA, Jones-Farmer LA, Champ CW, Woodall WH (2006) Effects of parameters estimation on control chart properties: a literature review. J Qual Technol 38(4):349–364Google Scholar
  10. Khoo MBC, Teoh WL, Castagliola P, Lee MH (2013) Optimal designs of the double sampling \(\bar{{X}}\) chart with estimated parameters. Int J Prod Econ 144(1):345–357CrossRefGoogle Scholar
  11. Maravelakis PE, Castagliola P (2009) An EWMA chart for monitoring the process standard deviation when parameters are estimated. Comput Stat Data Anal 53(7):2653–2664MathSciNetCrossRefMATHGoogle Scholar
  12. Montgomery DC (2005) Introduction to statistical quality control. Wiley, New YorkMATHGoogle Scholar
  13. Schoonhoven M, Riaz M, Does RJMM (2011) Design and analysis of control charts for standard deviation with estimated parameters. J Qual Technol 43(4):307–333Google Scholar
  14. Teoh WL, Khoo MBC, Castagliola P, Chakraborti S (2014) Optimal design of the double sampling chart with estimated parameters based on median run length. Comput Ind Eng 67(1):104–115CrossRefGoogle Scholar
  15. Wu Z, Ou Y, Castagliola P, Khoo MBC (2010) A combined synthetic & X chart for monitoring the process mean. Int J Prod Res 48(24):7423–7436CrossRefMATHGoogle Scholar
  16. Wu Z, Spedding TA (2000) A synthetic control chart for detecting small shifts in the process mean. J Qual Technol 32(1):32–38Google Scholar
  17. Zhang Y, Castagliola P, Wu Z, Khoo MBC (2011) The synthetic chart with estimated parameters. IIE Trans 43(9):676–687CrossRefGoogle Scholar
  18. Zhang Y, Castagliola P, Wu Z, Khoo MBC (2012) The variable sampling interval \(\bar{{X}}\) chart with estimated parameters. Qual Reliab Eng Int 28(1):19–34CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • H. W. You
    • 1
  • Michael B. C. Khoo
    • 1
  • P. Castagliola
    • 2
  • Yanjing Ou
    • 3
  1. 1.School of Mathematical SciencesUniversiti Sains MalaysiaPenangMalaysia
  2. 2.LUNAM Université, Université de Nantes & IRCCyN UMR CNRS 6597NantesFrance
  3. 3.Singapore Institute of Manufacturing Technology (SIMTech)SingaporeSingapore

Personalised recommendations