CORRECTIONS FOR ESTIMATED XBAR CONTROL CHARTS

XBAR control charts with estimated mean and standard deviation do not behave as those with known parameters and overestimate the probability of false alarm (PFA). We provide corrections to these charts so that they give the same PFA as the charts with known parameters. This correction is applied to the conventional constant "3" in the control limits. We consider eight different estimators for the process standard deviation. We develop a PFA formula for each estimator, using either a normal approximation or an exact t-distribution result, when available. The estimators are ranked in terms of their capability to reproduce the desired PFA, indicating the best and the worst ones in the group. This will help practitioners to choose a proper estimator for standard deviation. The PFA formulas are then inverted and solved for the constant "a" to replace "3" to reproduce the desired PFA. Formulas and tables for a are given which can easily be employed by the practitioners.