Quantities: Tolerance, DNS, Sigma(X), s

The distance to the nearer specification, DNS, is the distance from the mean to the nearer specification limit. It serves as a measure of how centered a process is within the tolerance. When the mean is centered at the target, the distance from the mean to each specification limit will be similar (converge). When the mean is biased toward one of the specification limits, the distance from the mean to each specification limit will diverge.

To calculate the DNS, subtract the Lower Specification Limit (LSL) from the mean and subtract the mean from the Upper Specification Limit (USL). The DNS will be the minimum of the two values. In practice, calculating the DNS is achieved using a MIN() function. This function returns the smaller of the two arguments. The formula for the DNS is written as:

When the mean falls inside the specification limits the values returned by the MIN() function will be positive. When the mean is less than the LSL, the Mean - LSL term of the MIN() function will be negative. When the mean is greater than the USL, the USL - Mean term will be negative.

The DNS will always be less than or equal to one-half of the tolerance. Thus, the relationship between the tolerance and the DNS is written as:

Rewriting this equation in terms of the tolerance yields the formula:

The global standard deviation, s, is the measure of dispersion taught in every introductory statistics course. By way of its calculation, it assumes that a dataset “can be logically considered to be one large homogeneous collection of values, all obtained from the same underlying and unchanging process” (Wheeler, Making Sense of Data, 162). Thus, when a process is operated predictably, the standard deviation summarizes the space used by the process in the past. However, when a process is operated unpredictably, the standard deviation tends to be inflated. In these instances, the standard deviation describes the past but this description provides no insight into whether or not the process is being operated predictably.

By understanding the quantities that compose the process capability indices, what each index measures becomes clear. The capability ratio, Cp, divides the tolerance by the quantity 6 Sigma(X). Thus, it expresses the space available to a process, allotted by the tolerance, as a multiple of the space required when the process is operated on-target and predictably.

The centered capability ratio, Cpk, divides the DNS by the quantity 3 Sigma(X).  Thus, it expresses the effective space available to a process as a multiple of the space required when the process is operated on-target and predictably. By comparing the centered capability ratio with the capability ratio, a characterization of how far off-target a process is operating is revealed. As the two ratios converge, the closer a process is operating to on-target. As the two ratios diverge, the further a process is operating from on-target.

Figure 1. As Cp and Cpk converge, the closer to on-target a process is operating. As they diverge, the further from on-target a process is operating.

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 centered performance ratio, Ppk, divides the DNS by the quantity 3s. Thus, it expresses the effective space available to a process as a multiple of the space used by the process in the past. By comparing the centered performance ratio with the performance ratio, a characterization of how far off-target a process is operating is revealed. As the two ratios converge, the closer a process is operating to on-target. As the two ratios diverge, the further a process is operating from on-target. Note that the characterization of how close to target a process is operating by comparing the performance ratios is the same characterization that is revealed by comparing the capability ratios.

Figure 2. As Pp and Ppk converge, the closer to on-target a process is operating. As they diverge, the further from on-target a process is operating.

Comparing the four indices

Comparing the four process capability indices to each other reveals insights into the behavior of the underlying causal system. When the values of the four indices diverge, it implies that the underlying causal system is influenced by both common causes of routine variation and assignable causes of exceptional variation. Thus, it is being operated unpredictably. When the values of the four indices converge (they will rarely be exactly the same), it implies that the underlying causal system is influenced by only common causes of routine variation. Thus, it is being operated unpredictably. The only way to definitely know what types of variation influence process behavior is by characterizing the underlying causal system using a process behavior chart.

Figure 3. When the process capability indices converge, it implies that the underlying causal system is behaving predictably. When the indices diverge, it implies that the underlying causal system is behaving unpredictably.