The Difference Between Cpk and Ppk

      What is the difference between Cpk and Ppk? How should they be properly used for process control and improvement? These are probably questions confusing many SPC users. This article will try to answer these questions.

      First, let’s compare the difference between Cpk and Ppk. Cpk is an indicator for the capability of the process, the best that the process can achieve, while Ppk is an indicator for the actual performance of the process. A process with, say, Cpk=1.67 may not show its full capability and only shows a Ppk below 1.67, say, 1.50. An analogy to this situation is a student who’s really smart and has the ability to become the top 1 student in the class (his capability), but it’s not necessary that he’ll become the top 1 student (his performance), for reasons such as less devotion than other students.

      Second, let’s see why Cpk is an indicator of the process capability, while Ppk is an indicator of the process performance.

      To answer this question, we must keep it mind that the first step for SPC control is to adjust the process and ensure it is statistically controlled. Meanwhile, for the sake of easy formulation, let’s assume that the process average coincides with the process target. In such a case

      In the above Equation I, (USL-LSL) is the accepted tolerance of the process, and σc the standard deviation of the measured samples (please be noted that it is the standard deviation of all the measured samples, not the average of the subgroups), an indicator of the total process variation. The higher their ratio (i.e. Cpk) is, the lower the chance is to produce a nonconforming product. For a process with a normal distribution and with the process target coinciding with the process average, the correspondence between Cpk and the defective rate (in terms of PPM) is given the following table:

      From the table, it can be seen that a higher Cpk results in a lower defective rate, indicating a better ability of the process in producing conforming products.

      In Equation I, σc is calculated with the following equation:

where Rbar is the average of the ranges of individual subgroups. Combining Equation I and Equation II, one can see that the capability of a process Cpk is determined by variation within each subgroup.

      At this point, one may wonder why the total process variation σc is only determined by the within-subgroup variation. How about the between-subgroup variation? To understand that, we must mention the precondition to evaluate the process capability again: the process is statistically controlled. For such a process, the between-subgroup variation is negligible (according to P.131 in the second revision of AIAG SPC’s manual, the between-subgroup variation is 0 for a statistically controlled process). Therefore, the total process variation, and hence Cpk, is only determined by the within-subgroup variation (please refer to the below image).

      In real life, however, a process cannot always be adjusted to statistically controlled state even there is no special cause detected in the control chart. Also, even if the process is in statistical control at the moment of evaluation, a special cause can occur at any future moment and take the process away from statistically controlled state. Therefore, for an actual process, its performance hardly shows its capability as indicated by Cpk. The total variation of the actual process consists of not only the within-subgroup variation, but also the between subgroup variation. The performance of the process, Ppk, is calculated with the following equation

where σp is total variation of the actual process, and it is the combined result of within and between-subgroup variation (please refer to the image below), and is calculated with the following equation:

where Xi is the data point and n is the total number of the sampled data.

      As a summary, Cpk is calculated for a process which is under statistical control. The between-subgroup variation is 0, and the process variation comes from within-subgroup variation only. The process in such state is the best performance that a process can achieve, so Cpk represents the process capability. On the other hand, Ppk is calculated for actual process which is not necessarily under statistical control. The process variation consists of both within and between subgroup variations. Therefore, Ppk reflects the actual performance of the process. Ppk can never be larger than Cpk.

      Finally, after understanding the difference of Cpk and Ppk, let’s consider how these two indicators should be used for process control and improvement. There’s no question that these two indicators should be monitored to meet customer or internal requriements (e.g. Cpk >1.67). Meanwhile, these two indicators should be monitored together and their difference is evaluated. A big difference between Cpk and Ppk indicates large between-subgroup variation, meaning special cause may exist in the process and actions should be taken to remove the special cause. Cpk and Ppk should never walk alone.