Capability analysis is an essential tool for statistical process control (SPC) and process improvement, applicable to a variety of environments from hospitals and labs to assembly and manufacturing. The formula for calculating a *C*_{pk} index is easily found:

* *

USL, upper specification limit; LSL, lower specification limit.

*Estimated sigma = average range/d2

Common understanding includes the fact that *C*_{pk} climbs as a process improves – the higher the *C*_{pk}, the better the product or process. Using the formula above, it’s easy to calculate *C*_{pk} once the mean, standard deviation, and upper and lower specification limits are known.

**Only One Specification or Tolerance**

But what if you have only one specification or tolerance – for example, an upper, but no lower, tolerance? Is *C*_{pk} advisable under these circumstances?

When faced with a missing specification, consider one of the following three options:

- Not calculating
*C*_{pk}, since all the variables are not known - Entering an arbitrary specification
- Ignoring the missing specification and calculating
*C*_{pk}on the only Z-value available

**Example: Plastic Pellet Manufacturer**

Examining a specific situation may clarify the outcome of each of these possibilities. A customer of a plastic pellet manufacturer has specified that the pellets should have a low amount of moisture content. The lower the moisture content, the better, but no more than 0.5 units is allowed; too much moisture will create manufacturing problems for the customer. The process is in statistical control.

This customer would undoubtedly not be satisfied with option 1 as *C*_{pk} has been specifically requested. With option 2, it could be argued that the LSL is 0, since moisture levels below zero are impossible. With a USL at 0.5 and LSL at 0, the *C*_{pk} calculation would be as follows.

Assume the X-bar = 0.0025 and estimated sigma is 0.15.

The customer is not likely to be satisfied with a *C*_{pk} of 0.005, and that number does not represent the process capability accurately.

Option 3 assumes that the lower specification is missing. Without an LSL, Z_{lower} is missing or nonexistent. Z_{min} becomes Z_{upper} and *C*_{pk} becomes Z_{upper} / 3.

Z_{upper} = 3.316 (from above)

*C*_{pk} = 3.316 / 3 = 1.10

A *C*_{pk} of 1.10 is more realistic than one of 0.005 for the data given in this example, and is more representative of the process itself.

**A Misleading ***C*_{pk}

*C*

_{pk}

As the example demonstrates, setting the lower specification to 0 results in a lower, misleading *C*_{pk}. In fact, as the process improves (here, moisture content decreases), the *C*_{pk} should have increased. If 0 was used as the LSL, however, the *C*_{pk} would have decreased. This is one clue that entering an arbitrary specification is not advised.

**Summary**

“What should be done when only one specification exists?” The (only) specification you have should be used, and the other specification should be left out of consideration or treated as missing. In these cases, use only Z_{upper} or Z_{lower}.

*C*_{pk} can and should be calculated when only one specification exists, provided only the remaining valid specification is used. As the example demonstrates, the missing specification should remain missing and not be artificially inserted into the calculation.

Zero in this case is not a specification limit but a process boundary and therefore should not be used in the calculation of a parameter that calls for a specification.

This is a common error many do not quite grasp. I have seen it too in a slight different situation like the specification of max bow -let’s call it b- in a bimetal that is supposed to be flat. In this case in this case if the specification is interpreted to be 0 to b (positioning the bimetal as an inverted U on the table and it always bends to the same side so there is no -b measures) the closer the part is to flat the worst the Cpk, unless using the unilateral b USL where farther away from b and the smaller the deviation will translate into a flatter more consistent part.