Control Charts Applicable When the Fraction Nonconforming is Small

Abstract

It has been proposed that when the proportion of nonconforming product is extremely small that one base 3σ control charts on the 0.2777th power of X, the number of items sampled until a nonconforming item is found. This choice of exponent renders the transformed variable approximately normal in distribution. In this paper, that proposal is compared to the alternative of using a logarithmic transformation of X as the basis for 3σ control charts. Expressions are derived for the OC curves of both procedures. It is shown further that if probability limits are used in lieu of 3σ limits, the OC curves of the two procedures are identical. If probability limits are used, the power transformation is preferred to the log transform, since of the two, its distribution is more nearly normal and, thus, more closely justifies the use of the supplementary tests commonly used to enhance the power of control charts. Computation of the average run lengths shows that the power transformation gives poor protection against worsening quality and that the log transformation results in an unacceptably short in-control ARL value. It is shown that either transformed value gives greatly improved performance when used in an EWMA scheme. The power transformation is recommended over the log transformation as the basis for an EWMA chart because it will detect an increase in the fraction nonconforming of up to a factor of three with a smaller resultant ARL. Moreover, it will detect any decrease in the fraction nonconforming more quickly.