We discuss the estimation of a process capability index for three-dimensional data. Initially, we focus on the case in which the engineering tolerance associated with the measurements is a sphere. Then, we extend the discussion to the more general case in which the engineering tolerance is ellipsoidal. In both cases, we develop summary measures for repeatability and reproducibility, to be used in the context of a process capability index.
In the spherical tolerance case we define summary measures, where each measure is based on the diameter of a sphere that leads to a pre-specified capture rate (we will use here 99%). As a process capability index, we 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 will be 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 (again, we will use here 99%). As a process capability index, we propose ratios, where each ratio is the major axis of such ellipsoid divided by the major axis of the tolerance ellipsoid.
We present two algorithms in the language R aimed at facilitating the estimation of our summary measure of variability. The first algorithm evaluates the probability that a linear combination of three (or fewer) independent chi-square variables will be less than or equal to a given constant. The second algorithm estimates the value a linear combination of chisquare variables is less than or equal to, given a pre-specified probability. In addition, we offer an algorithm in the language R for computing the process capability index in the context of color metrics.
We present applications to color measurements and to R&R analysis of color metrics.
We show how the components of variance in these three-dimensional measurements can be easily compared to each other and to the tolerance region, using the single-dimensional summary measures of process capability.
Library of Congress Subject Headings
Process control--Statistical methods; Tolerance (Engineering); Elliptic functions
Applied Statistics (MS)
Department, Program, or Center
The John D. Hromi Center for Quality and Applied Statistics (KGCOE)
Joseph G. Voelkel
Daniel R. Lawrence
Fotak, Veljko, "A Process Capability Index for Three-Dimensional Data with Circular or Ellipsoidal Tolerances." (2005). Thesis. Rochester Institute of Technology. Accessed from
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