A Process Capability Index for Three-Dimensional Data

I discuss the estimation of a process capability index for three-dimensional data. Initially, I focus on the case in which the engineering tolerance associated with the measurements is a sphere. Then, I extend the discussion to the more general case in which the engineering tolerance is ellipsoidal. In both cases, I develop summary measures for repeatability and reproducibility, to be used in the context of a process capability index. In the spherical tolerance case I define summary measures, where each measure is based on the diameter of a sphere that leads to a pre-specified capture rate. As a process capability index, I propose ratios, where each ratio is the diameter of such a sphere divided by the diameter of the tolerance sphere. In the ellipsoidal tolerance case, such summary measure is based on the length of the major axes of the ellipsoid of identical shape and orientation to the tolerance ellipsoid providing a pre-specified capture rate. As a process capability index, I propose ratios, where each ratio is the major axis of such ellipsoid divided by the major axis of the tolerance ellipsoid.