"What is the correlation between Cpk and a control chart's UCL and LCL values?"
I belong to several groups on the Linked In web site and find many of the group discussions both interesting and informative. Recently, a member of the American Society for Quality (ASQ) group posed the question, "I'm new in the quality area and eager to go deeper into it. Do help me. What is the correlation between Cpk and UCL, LCL value?"
Several group members – most of them Six Sigma Black Belts – replied; but none of them answered the young engineer's question. Instead, they recommended a youtube video, CQE test preparation book ("See item 5 on page 34"), several on-line articles, and a couple of other sources. I decided to pitch in and try to answer to original inquiry, as follows:
There is no relationship between the process capability index (Cpk) and the UCL and LCL values on a control chart. Upper and lower control limits (UCL and LCL) are used to determine if the process is stable or not. If the process is not stable, you'll find points plotting outside the control limits and/or a non-random pattern of points within the control limits. It will have no capability; and any Cpk calculated for an out-of-control process is invalid.
On the other hand, if all data plot in a random pattern within the UCL and LCL, we conclude that the process is stable – under the influence of common causes of variation only. This indicates that the process does have a capability, which is defined as the extent of the random, inherent, common cause variation we observe. The UCL and LCL have done their job; but their actual, calculated values have nothing to do with the Cpk calculation. Proceed to calculate the Cpk for the stable process using the standard formulas for Cpkl and Cpku – neither of which include the UCL and LCL values.