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Designing Phase I ―X Charts with Small Sample Sizes
Author(s) -
Champ Charles W.,
Jones L. Allison
Publication year - 2004
Publication title -
quality and reliability engineering international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 62
eISSN - 1099-1638
pISSN - 0748-8017
DOI - 10.1002/qre.662
Subject(s) - standard deviation , estimator , statistics , mathematics , multivariate normal distribution , multivariate statistics , sample size determination , joint probability distribution , chart , x bar chart , normal distribution , control chart , control limits , computer science , process (computing) , operating system
Methods are examined for obtaining probability limits for Phase I Shewhart ―X charts when the process mean and standard deviation are estimated. The design methods assume m independent random samples of size n will be taken periodically from a normally distributed process. It is shown that the joint distribution of the standardized subgroup means follows either an approximate or exact multivariate t distribution, depending on the standard deviation estimator used. The multivariate t distribution is used to define the probability limits for the Phase I ―X chart. Extensive simulation compares the performance of the proposed limits with other design procedures. Tables of design constants are given for $m<20$ , and simple procedures for obtaining design constants are given for $m\geq 20$ . Copyright © 2004 John Wiley & Sons, Ltd.