Centered Capability Ratio, Cpk
The Centered Capability Ratio, Cp, compares the effective space available to a process with the space required by a process. Unlike the space available, the effective space available considers how well the available space is being used by integrating the mean of the process data into its calculation.
The space required is denoted by the term 3 Sigma(X), where Sigma(X) is a generic placeholder for one of several within-subgroup measures of dispersion. The effective space available describes the location of the process mean relative to the distance to the nearer specification or DNS. The formula for the Centered Capability Ratio is:
The relationship between the effective space available and the space required is further clarified in Figure 1. Here, a generic distribution of individual values is shown in the context of the specification limits (the space available), the effective space available (2 DNS), and the process limits (the space required). Note that the distribution in Figure 1 is bias toward the Lower Specification Limit (LSL), not centered. This bias towards the LSL is as if the process is operating within specification limits that have a width equal to 2 DNS instead of the space available.
When a process is operated in the middle of the specification limits (on-target), the Centered Capability Ratio (Cpk) and the Capability Ratio (Cp) will converge (rarely will the values of Cp and Cpk be the same). As the process mean deviates from the target, the Centered Capability Ratio (Cpk) will be smaller than the Capability Ratio (Cp). Thus, the value of the Centered Capability Ratio relative to the Capability Ratio provides some insight into how close to target a process is operating. As the values converge, the closer to on-target the process is operating. As the values diverge, the further from on-target the process is operating.
A word of warning
The Capability Ratio, like all process parameters, is not well-defined until the underlying causal system is operated predictably. Whenever a process is operated unpredictably, that is it is influenced by both common causes of routine variation and assignable causes of exceptional variation, the process parameters, including the process mean, the process standard deviation, and the process capability indices, will change over time. While we can always compute statistics regardless of the characterization (predictable or unpredictable), the only time this arithmetic can be used to estimate process parameters is when the process is operated predictably. When a process is operated unpredictably “the process parameters are changing and are therefore divorced from the statistics” used to calculate them. (Wheeler, More Capability Confusion, (Quality Digest, May 2017), 2).