Abstract
The run length distribution of\(\bar X\) charts with unknown process variance is analized using numerical integration. Both traditional\(\bar X\) chart limits and a method due to Hillier are considered. It is shown that setting control limits based on the pooled standard deviation, as opposed to the average sample standard deviation, provides better run length performance due to its smaller mean square error. The effect of an unknown process variance is shown to increase the area under both tails of the run length distribution. If Hillier’s method is used instead, only the right tail of the run length distribution is increased. Collani’s model for the economic design of\(\bar X\) charts is extended to the case of unknown process variance by writing his standardized objective function in terms of average run lengths.
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7 References
Collani Ev (1988) A unified approach to optimal process control. Metrika 35:145–159
Collani Ev (1989) Economically optimalc andnp charts. Metrika 36:215–232
Del Castillo E, Montgomery DC (1994) Short-run statistical process control: Q-chart enhancements and alternative methods. Quality and Reliability Engineering International 10:87–97
Ghosh BK, Reynolds MR, Van Hui Y (1981) Shewhart\(\bar X\) charts with estimated variance. Communications in Statistics, Theory and Methods 18:1797–1822
Hillier FS (1964)\(\bar X\) control limits based on a small number of subgroups. Industrial Quality Control 24–29
Hillier FS (1967) Small sample probability limits for the range chart. Journal of the American Statistical Association 62:1488–1493
Lorenzen TJ, Vance LC (1986) The economic design of control charts: A unified approach. Technometrics 28:3–10
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in C. New York Cambridge University Press
Quesenberry CP (1991) SPC Q charts for start-up processes and short or long runs. Journal of Quality Technology 23:231–224
Quesenberry CP (1993) The effect of sample size on estimated limits for\(\bar X\) andX control charts. Journal of Quality Technology 25:237–247
Wasserman GS, Sudjianto A (1993) Short run SPC based upon the second order dynamic linear model for trend detection. Communications in Statistics, Simulation and Computation 22:1011–1036
Yang CH, Hillier FS (1970) Mean and variance control charts based on a small number of subgroups. Journal of Quality Technology 2:9–16
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Del Castillo, E. Run length distributions and economic design of\(\bar X\) charts with unknown process variance. Metrika 43, 189–201 (1996). https://doi.org/10.1007/BF02613907
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DOI: https://doi.org/10.1007/BF02613907