Designing Phase I X̄ charts with small sample sizes

Methods are examined for obtaining probability limits for Phase I Shewhart X̄ 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 X̄ 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 ≥ 20.