In my previous blog,Why Individuals control charts are typically better than x-bar and R charts , the advantages of a 30,000-foot-level reporting methodology was described. It was shown in this earlier blog how 30,000-foot-level control charting techniques could be used to determine if a process were stable. It was also stated that if a process had a recent region of stability, a process could be proclaimed predictable; i.e., unless either a positive or negative change occurred in the process, one would expect to produce about the same result as its current performance.
For regions of stability, process capability indices can be used to describe how the process is performing relative to specification limits. However, there are issues with traditional process capability indices reporting; i.e., use of Cp, Cpk, Pp, and Ppk. This blog will address these issues. My next blog will provide an improved process capability reporting methodology that is easier to interpret and can be used throughout an organization when reporting process capability/performance.
The process capability index Cp represents the allowable tolerance interval spread in relation to the actual spread of the data when the data follow a normal distribution. This equation is:
where USL and LSL are the upper specification limit and lower specification limit, respectively, and 6s describes the range or spread of the process; i.e., 6 times standard deviation. Data centering is not taken into account in this equation.
Cp addresses only the spread of the process; Cpk is used concurrently to consider the spread and mean shift of the process. Mathematically, Cpk can be represented as the minimum value of the two quantities:
In these Cp and Cpk relationships, the standard deviation (σ) variable is to be "short term." The process capability/performance indices Pp and Ppk have a similar relationship; however, the standard deviation value is to be "long term" for these calculations.
If two people were to examine the same process during the same time frame, would you expect that these two people would get about the same values for Cp, Cpk, Pp, and Ppk process capability indices? Not necessarily. For example, if one person chose an x-bar and r chart’s data to determine process capability, he could get a very different response than someone who chose an individuals chart.
To illustrate this point, let’s consider the data presented in Table 1, which are the same data that were used to create the control charts in a previous blog. For this type of data, a traditional control chart selection approach would lead to an x-bar and R chart of all the data; however, someone could have chosen to collect only one sample from each time period. For a data set to describe this scenario, I will later use the first column of this data set.
In a previous blog, the x-bar and R chart was considered out of control, while the individuals control chart of the data from the first column was considered in control. A previous blog elaborated on why this occurred.
For this process, we will be focusing on process capability statements. Process capability statements should not be made for processes that are out of control, but this point, for this data set, will be addressed in a future blog.
A process capability analysis of the x-bar and R data is shown in Figure 1, while Figure 2 does the same for the individuals data set.
From the output of Figures 1 and 2, we noticed a very different response for Cp and Cpk, while there was not much difference between Pp and Ppk:
The reason for the difference in calculated Cp and Cpk values is because of the standard deviation value that was used in the calculations.
When Cp and Cpk are determined from an x-bar and R chart, process-capability standard deviation is calculated from the relationship:
where R-bar is the average of the response range within subgroups and d2 is a table constant; e.g., d2 equals 2.326 for a five subgroup size.
When Cp and Cpk are determined from an individuals control chart, standard deviation is calculated from the relationship:
where MR is the moving range between adjacent subgroups.
Basically, the individuals control chart considers variability between subgroups in the determination of Cp and Cpk, while the x-bar and R control chart doesn’t. For the reasons described in a previous blog, most situations should consider variability between subgroups when calculating control limits. A similar position should also be taken for the calculation of standard deviation when determining process capability indices; i.e., variability between subgroups needs to be considered when making capability/performance statements. Hence, an individuals control chart calculation approach would be preferable over an x-bar and R chart approach.
However, requiring an individuals control chart for these assessments does not solve all the issues with process capability indices reporting. For example, the frequency of subgrouping can also significantly impact the value for a calculated process capability response; i.e., a frequently sampled process could have a much smaller between-subgroup moving range (standard deviation) than one that was sampled infrequently.
Pp and Ppk has fewer issues than Cp and Cpk; however, the reporting of Pp and Ppk indices can still cause much confusion relative to the physical interpretation of reported values.
My next blog will describe an alternative approach to process capability reporting that not only addresses the above issues but also provides a statement that is easier to understand and convey to others.
Reference: The content of this blog was taken from Chapter 11 of IEE Volume III .