True Position Capability

Some manufacturing processes, and the allowable tolerances that define acceptable quality, can best be summarized by the metaphor of "hitting a target." For example, when drilling holes at specific locations, the quality requirement is best defined by circles around the desired locations; every time that a hole is drilled outside the acceptable quality (circle), the respective part is rejected.

Tolerances (specification limits) defined as a circle in the two-dimensional plane are also called positional tolerances. For such processes, the standard process capability values (ratios) are not appropriate, because while the process may be within-specs on each individual dimension, the respective point (in the two-dimensional plane) may be unacceptably far away from the desired goal (point).

Specifically, the standard potential capability index for a process in a single dimension is defined as Cp=(USL-LSL)/(6*Sigma). By taking the radius of a circle as 1/2 of the quantities defined in the numerator and denominator of this equation, Positional Cp can be computed as:

where D is the tolerance diameter.

Positional Cpk is computed as:

Note that in these computations it is assumed that the specification limits in both dimensions are the same (and define a circle around the target), that the two variables X and Y (the two measured dimensions) are uncorrelated, and that the variability in each dimension (for X and Y) is approximately the same. As a practical matter, the larger of the observed variances in the two dimensions is taken to estimate s.

For additional details, see also Krishnamoorthi, 1990.