Designing Phase I ―X Charts with Small Sample Sizes

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.