Performance Ratio, Pp
The performance ratio, Pp, divides the tolerance by the quantity 6s. Thus, it expresses the space available to a process, allotted by the tolerance, as a multiple of the space used by the process in the past. As the space used in the past and the space required by the process converge, so too will the performance ratio converge with the capability ratio. This convergence is indicative of a process that is being operated predictably.
The relationship between the space available and the space used in the past is clarified in Figure 1. Here, a generic distribution of individual values is shown in the context of the specification limits (the space available) and the space used in the past, as defined by the term Average ± 3s. The s in the term 6s is the global standard deviation statistic. It should not be confused with the within-subgroup measure of dispersion Sigma(X), the defining term in the denominator of the capability ratio and the centered capability ratio.
Figure 1. The relationship between the space available and the space used in the past by the process.
A word of warning
The performance 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, Making Sense of Data, 2).
It is for the above reasons that the process capability indices should always be reported with the additional context of a process behavior chart and a capability histogram. The process behavior chart will characterize process behavior as predictable or unpredictable and, in doing so, reveal the voice of the process. The capability histogram will visualize the relationship between the specification limits (tolerance) and the process data. In doing so, the capability histogram reveals the relationship between the process data and the voice of the process.
Figure 2. The X chart portion of an XmR chart (left) and the associated capability histogram (right).