Optimal Limits for Attributes Control Charts

Abstract

Attributes control charts have historically been used with 3-sigma limits. When such an approach is used there is the implicit assumption that the normal approximation to the binomial and Poisson distributions will be adequate. Control chart properties are determined by the reciprocals of the tail areas, but most approximations, including normal approximations, perform the poorest in the tails of a distribution. Normal approximations can also be poor when the binomial and Poisson parameters are small, as will occur frequently in applications. Previously proposed approaches include using a Q chart or using either an arcsin transformation for binomial data or a square root transformation for Poisson data. Although being superior to a normal approximation, these alternative approaches will not necessarily produce tail areas that are close to the nominal tail areas. In this paper we provide tables that can be used to produce optimal control limits for u and c charts, and regression is used in conjunction with the optimal limits to produce approximations that can be used for interpolating between the tabular values. The regression equations are then suitably adapted for use with p and np charts, for which a complete set of tables of optimal control limits would not be practical to construct.

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