A Simple Guide to Between / Within Capability

Having delivered training courses on capability analyses with Minitab, several times, I have noticed that one question you can be absolutely sure will be asked, during the course, is: What is the difference between the C_pk and the P_pk indices?

P_pk vs. C_pk indices

The terms C_pk and P_pk are often confused, so that when quality or process engineers refer to the C_pk index, they often actually intend to mean P_pk indices.

P_pk is used to assess the long-term, overall variability, whereas C_pk is the capability index for short-term, potential variability. In this blog post, I will try to make this difference more explicit.

Consider the graph below. Suppose that during a full week period, measurements have been collected day after day. Suppose also that the process we are monitoring is cyclical. The amount of variability within one day is quite small, but because of the cyclical behavior or process instability from day to day, the overall variability during the whole week is much larger than the variability within single days. The P_pk is estimated from the dispersion of all individual values during the whole period, whereas the C_pk is based on variations only within subgroups (within days in my example).

Customers are likely to be affected by the overall variability, and therefore only the P_pk index matters to them. The C_pk often provides an overly optimistic capability estimate because cyclical behaviors between days are not taken into consideration.

As far as the vendor is concerned, it is useful to know that potentially, if the cyclical, unstable behavior was successfully dealt with, the P_pk would be improved and would become equivalent to the C_pk. The graph below illustrates a situation in which the P_pk and the C_pk indices are equivalent because the process is stable. In this case, variations within subgroups are similar to variations during the whole period.

Between / Within capability

The sources of variability that affect processes in the long term might be different from the ones that take place in the short run. For example, within-batch variability is often much smaller than variations between batches, since parts in the same batch are often processed on the same tools within a short period of time, and have the same processing history. Other sources of variations—such as seasonal changes and modifications in the environment—may have a longer-term impact.

The Between/Within capability method can be used to estimate short-term variability even more accurately than described above. The variability within batches is still used to estimate short-term variability, but as far as the differences between batches are concerned the approach is slightly more subtle. One part of between-batches variability will be accounted for as short-term variability, by considering only differences between consecutive batches (in time order).

Referring to my previous example, Within variability represents within-batch variability, whereas Between variability represents short-term fluctuations between batches. To estimate these short-term variations between batches, only differences between the averages of consecutive batches are considered (Moving Ranges between averages of consecutive batches). Within and Between variations estimates are then compounded together to calculate a short-term between/within C_pk, whereas the P_pk is still based on the overall long-term variability, considering all individual values (not in time order) during the whole period.

In the graph below, the process is cyclical in nature and a long-term trend is clearly visible, although differences between averages of consecutive batches are small (variability between consecutive batches) it is clear that the overall variability during the full period is much larger due to this long-term trend. Differences between consecutive batches fail to capture the full extent of the overall variability, but are good estimates of short-term variability.   

Conclusion

It is important to differentiate short-term variability from long-term variability, because if a process is affected by drifts and systematic long term trends, it will become unstable and therefore unpredictable. A P_pk that is estimated today may not be valid tomorrow because of a long term process shift. Process stability and fluctuations due to random / common causes are necessary to ensure a predictable behavior. The P_pk (overall capability) index should therefore be as close as possible to the C_pk (short term) estimate.