Kiani, Mehdi and Panaretos, John and Psarakis, Stelios (2008): A new procedure for monitoring the range and standard deviation of a quality characteristic. Forthcoming in: Quality and Quantity

PDF
MPRA_paper_9067.pdf Download (336kB)  Preview 
Abstract
The Shewhart and the Bonferroniadjustment R and S chart are usually applied to monitor the range and the standard deviation of a quality characteristic. These charts are used to recognize the process variability of a quality characteristic. The control limits of these charts are constructed on the assumption that the population follows approximately the normal distribution with the standard deviation parameter known or unknown. In this article, we establish two new charts based approximately on the normal distribution. The constant values needed to construct the new control limits are dependent on the sample group size (k) and the sample subgroup size (n). Additionally, the unknown standard deviation for the proposed approaches is estimated by a uniformly minimum variance unbiased estimator (UMVUE). This estimator has variance less than that of the estimator used in the Shewhart and Bonferroni approach. The proposed approaches in the case of the unknown standard deviation, give outofcontrol average run length slightly less than the Shewhart approach and considerably less than the Bonferroniadjustment approach.
Item Type:  MPRA Paper 

Original Title:  A new procedure for monitoring the range and standard deviation of a quality characteristic 
Language:  English 
Keywords:  Shewhart, Bonferroniadjustment, Average run length, R chart, S chart 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General 
Item ID:  9067 
Depositing User:  J Panaretos 
Date Deposited:  11. Jun 2008 07:29 
Last Modified:  11. Jan 2014 02:49 
References:  Arnholt, A. T., and Hebert, J. L. Estimating the Mean With Known Coefficient of Variation. The American Statistician, Vol. 49, No. 4, pp. 367369 (1995). Donatos, G. S.. A Monte Carlo Study of $k$Class Estimators for Small Samples with Normal and NonNormal Disturbances. The Statistician, Vol. 38, No. 1, pp. 1120 (1989). Glasser, G. J.. On Estimators for Variances and Covariances. Biometrika, Vol. 49, No. ½, pp. 259262 (1962). Gurland, J., and Tripathi, R. C.. A Simple Approximation for Unbiased of the Standard Deviation. The American Statistician, Vol. 25, No. 4, pp. 3032 (1971). Healy, M. J. R.. A Mean Difference Estimator of Standard Deviation in Symmetrically Censored Normal Samples. Biometrika, Vol. 65, No. 3, pp. 643646 (1978). Johnson, N. L., Kotz, S., and Balakrishnan, N.. Continuous Univariate Distributions. 2nd edition. Wiley, New York (1994). Khan, R. A. A Note on Estimating the Mean of a Normal Distribution with Known Coefficient of Variation. Journal of the American Statistical Association, Vol. 63, No. 323, pp. 10391041 (1968). Markowitz, E. Minimum MeanSquareError Estimation of the Standard Deviation of the Normal Distribution. The American Statistician, Vol. 22, No. 3, pp. 2626 (1968). Montgomery, D. C. Introduction to Statistical Quality Control. 4th edition. Wiley, New York (2001). Nedumaran, G., and Pignatiello, J. J. On Constructing Retrospective Xbar Control Chart Limits. Quality and Reliability Engineering International, Vol. 21, pp. 8189 (2005). Ott, E. R. Process Quality Control. McGrawHill Book Company, New York (1975). Prescott, P. Use of a Simple RangeType Estimator of standard deviation in Test of Hypotheses. Biometrika, Vol. 58, No. 2, pp. 333340 (1971a). Prescott, P.. Distribution of the Normal Scores Estimator of the Standard Deviation of a Normal Population. Biometrika, Vol. 58, No. 3, pp. 631636 (1971b). Quesenberry, C. P. SPC Methods for Quality Improvement. John Wiley & Sons (1997). Rohatgi, V. K. Statistical Inference. John Wiley & sons, New York (1984). Ryan, T. P. Statistical Methods for Quality Improvement. John Wiley & Sons (1989). Shewhart, W. A. Economic Control of Quality of Manufactured Product, D. van Nostrand Co., New York (1931). Smith. G. M. Statistical Process Control and Quality Improvement. PrenticeHall, Inc. (1998). Tsai, T. R., Lin, J. J., Wu, S. J., and Lin, H. C. On estimating Control Limits of Xbar Chart When the Number of Subgroups Is Small. Int J Adv Manuf Technol, Vol. 26, pp. 13121316 (2005). Vardeman, S. B. A Brief Tutorial on the Estimation of the Process Standard Deviation. IIE Transactions, Vol. 31, pp. 503507 (1999). Watson, S. Evaluation of Semivariance Estimators Under Normal Conditions. The Statistician, Vol. 46, No. 4, pp. 495503 (1997). 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/9067 