Capability Clarity
The only time the process capability indices make practical sense (that is to say they are well-defined and useful) is when a process has been shown to be operating in a state of statistical control. This is the state that is achieved when, by using a process behavior chart, future process behavior can be predicted within limits. That is, process behavior has been characterized as predictable. This occurs when all of the values on a process behavior chart fall within the process limits. Without this knowledge, the process capability indices may be misleading or altogether false indicators of what the process will do in the future.
The logic for using the process capability indices is outlined in the flowchart shown in Figure 1. Here, efforts always begin with characterization (to learn more about characterization click here). When a process is characterized as predictable, the logic on the left side of the flowchart is followed. In these instances, the underlying causal system is influenced by only common causes of routine variation. While many in number, common causes are minimal in their effect. Thus, a predictable process will change very little over time. Observations collected from a predictable process can be used to compute summary statistics that both describe properties of the data and estimate process parameters (like the process capability indices).
When a process is characterized as unpredictable, the logic on the right side of the flowchart is followed. In these instances, the underlying causal system is influenced by both common causes of routine variation and assignable causes of exceptional variation. While few in number, assignable causes are dominant in their effect. Thus, the underlying causal system will bounce and shift in unpredictable ways. Observations collected from an unpredictable process can be used to compute summary statistics but those summary statistics ONLY describe properties of the data. Attempts to use these statistics to estimate process parameters (like the process capability indices) are functionally meaningless. This is because the underlying causal system is always changing. The only time process parameters are well-defined is when a process is operated predictably.
Figure 1. Flowchart outlining the logic for when the process capability indices are reliable.
Although you may be required to report the process capability indices regardless of the characterization, the knowledge revealed by characterization is quintessential to how you move forward. When a process is operated unpredictably, any changes to the process that do not address the influence of assignable causes will be worthless. While few in number, assignable causes are dominant in their effect. The only way to improve an unpredictable process is to eliminate the influence of the assignable causes. The only way to eliminate the influence of assignable causes is to identify them with a process behavior chart. Thus, when required to report the process capability indices, always present them with the associated process behavior chart. This will give the indices visual context that is overlooked when the values alone are presented.
Figure 2. XmR chart of process data.
While process behavior charts reveal the types of variation that influence process behavior, they do not reveal how the process is operating with regard to specifications. Thus, in addition to a process behavior chart, process capability indices should also be presented with a capability histogram like the one shown in Figure 3. Capability histograms reveal the shape, center, and spread of process data in the context of the Upper Specification Limit (USL), the Lower Specification Limit (LSL), the target and the mean of the distribution. The specification limits define the voice of the customer. They define what a customer is willing to pay for and what they will accept.
Figure 3. Capability histogram, the associated process capability indices, and the characterization of the underlying causal system.
By presenting the process capability indices with the context of both a capability histogram and a process behavior chart when reporting, a complete picture of the process lifecycle is revealed. The process behavior chart reveals the voice of the process. The capability histogram reveals the voice of the customer. The process capability indices reveal the relationship between the voice of the process and the voice of the customer. Anything less is a failure of due diligence and understanding.
“Just because two quantities have meaning in their own right does not mean that you can form a ratio of those two quantities and end up with a meaningful result.”
— Donald J. Wheeler, More Capability Confusion, (Quality Digest, 2017)
Calculating capabilities: Equations & terms
Among the many acronyms and terms that torment the hallowed grounds of industry are the well meaning but insidious quartet known as the process capability indices. Composed of four ratios—the Capability Ratio (Cp), the Centered Capability Ratio (Cpk), the Performance Ratio (Pp), and the Centered Performance Ratio (Ppk)—the process capability indices have become a standard, albeit misunderstood, method by which processes are graded and their performance quantified.
The most nefarious misunderstanding that plagues this capability quartet is the confusion that surrounds the calculation of each index. Depending on the source, the formulas used to calculate the process capability indices assume some perplexing forms and integrate some perplexing terms. Thus, before any discussion about their utility or debate about their shortcomings ensues, an understanding of the process capability indices begins with a review of their mathematical forms. The formulas for each of the indices, as defined by the statistician and quality control expert Donald J. Wheeler throughout his extensive body of work are:
Reviewing these formulas reveals both similarities and differences. Where the numerators of the Capability Ratio and Performance Ratio are defined by the Tolerance, the numerators for the Centered Capability Ratio and Centered Performance Ratio are defined by the distance to the nearest specification, DNS. Where the denominators for the Capability Ratio and Centered Capability Ratio are defined by Sigma(X), a within-subgroup measure of dispersion, the denominators for the Performance Ratio and Centered Performance Ratio are defined by the global standard deviation statistic, s.
While straightforward when articulated in this context, confusion abounds with the terms that define the denominators for these ratios. The Sigma in Sigma(X) is reliably assumed to be the standard deviation statistic. This is the same confusion that plagues the unwary when calculating process limits. Shewhart’s generic formula for three-sigma limits is written as:
Here, Average is the mean of the dataset and Sigma is a generic placeholder for one of several within-subgroup measures of dispersion, NOT the standard deviation statistic. When the standard deviation is used to calculate process limits, the limits will be inflated. This yields process limits that are inflated and thus less sensitive to the influence of assignable causes of exceptional variation. This can result in misleading or altogether wrong characterizations of the underlying causal system (for details see BrokenQuality.com/how-to-calculate-process-limits). Assuming the Sigma(X) in the denominator of the Capability Ratio and Centered Capability Ratio yields values that are similarly misleading. To understand this a review of the terms that compose the the process capability indices is necessary.
Tolerance
Representing the voice of the customer, the Tolerance defines the space available to a process. When the tolerance is not provided, it is calculated by subtracting the Lower Specification Limit (LSL) from the Upper Specification Limit (USL).
DNS
The distance to the nearest specification limit (DNS) is the distance from a dataset’s mean to the nearest specification limit. The DNS serves as a measure of how centered a process is relative to the specification limits. When the process mean is centered at the target, the distance from the mean to the LSL and the distance from the mean to the USL will converge.
To calculate the DNS, subtract the LSL from the mean and subtract the mean from the USL. The DNS will be the minimum of these two values. In practice, this is achieved using a MIN() function. When proivided with two arguments, a MIN() function returns the smaller of the two values.
When the mean falls inside the specification limits the arguments inside the DNS formula will return positive values. When the mean is less than the LSL, the (Mean - LSL) term will be negative. When the mean is greater than the USL, the (USL - Mean) term will be negative.
Since the DNS will always be less than or equal to 1/2 of the tolerance the relationship between the tolerance and the DNS can be written as:
Solving this equation for the tolerance yields the formula:
Thus, the 2DNS term used in the Centered Capability Ratio (Cpk) and the Centered Performance Ratio (Ppk) emerges as a product of the relationship between the tolerance and the DNS.
Sigma(X)
Sigma(X) is a generic placeholder for one of several within-subgroup measures of dispersion that changes with subgroup size. When a dataset is composed of logically comparable individual values, Sigma(X) relates the average moving range to the bias correction factor, d2. The value of d2 will always be 1.128 when a dataset is composed of individual values. Sigma(X) is calculated by dividing the average moving range by d2 = 1.128.
Unfamiliar with the average moving range? Visit BrokenQuality.com/how-to-build-an-xmr-chart for details.
Standard deviation
The standard deviation statistic is the measure of dispersion taught in every introductory statistics course. Its calculation assumes that a dataset “can be logically considered to be one large homogenous collection of values, all obtained from the same underlying and unchanging process” (Making Sense of Data, Donald J. Wheeler, (SPC Press, 2003), p. 162.) Within the context of the process capability indices, the standard deviation describes the past process performance.
The process capability indices in practice
With an understanding of the terms that compose the process capability indices in hand, the next step is to put them to practice. Enter your email below to download Understanding the Process Capability Indices. This essay puts your new found understanding of the terms to practice. It discusses how the indices are calculated and what their calculation indicates about a process. It also explores how process behavior charts and capability histograms provide the additional context that is necessary to take actions that reduce costs and improve quality.
Figure 4. Cp is the space available (defined by the specification limits) divided by the space required (defined by the process limits).
The reader is also encouraged to explore the work of the statistician and quality control expert Donald J. Wheeler at spcress.com. Dr. Wheeler’s work is the foundation of the above discussion and the details outlined in Understanding the Process Capability Indices.
