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Introduction to Process Capability

Posted by isoeasy on May 10, 2006

Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification "width") to the spread of the process values, as measured by 6 process standard deviation units (the process "width").

Process Capability Indices
A process capability index uses both the process variability and the process specifications to determine whether the process is "capable" We are often required to compare the output of a stable process with the process specifications and make a statement about how well the process meets specification. To do this we compare the natural variability of a stable process with the process specification limits.

A capable process is one where almost all the measurements fall inside the specification limits.


* Cp is the simple process capability index. It is the process width divided by 6 times sigma, its estimated within-subgroup standard deviation, where the process width = Upper Spec Limit minus Lower Spec Limit. If Cp < 1, the process is wider than the spec limits, and is not capable of producing all in-specification products. Cp could be greater than one, but bad parts could still be being produced if the process is not centered. Thus, there is a need for a capability index which takes process centering into account: Cpk.

* Cpk is the difference between x double bar and the nearer spec limit divided by 3 times sigma. If Cpk >=1, then 99.7% of the products of the process will be within specification limits. If Cpk <1, then more non-conforming products are being made.

* Cu: The difference between the process mean and the upper spec limit, divided by 3 sigma, or 3 times the standard deviation.

* Cl: The difference between the process mean and the lower spec limit, divided by 3 sigma, or 3 times the standard deviation.

* Cpk: Cpk is the difference between the process mean and the nearer spec limit divided by 3 times sigma. (Cpk is the lesser of Cuand Cl.). If Cpk>=1, then at least 99.7% of all products of the process will be within specification limits. If Cpk<1, then some non-conforming products are being made, and you may need to study your process to see how it can be improved.

* Pp: The Pp index is used to summarize a system's performance in meeting two-sided specification limits (upper and lower). Like Ppk, it uses actual sigma (sigma of the individuals), and shows how the system is actually running when compared to the specifications. However, it ignores the process average and focuses on the spread. If the system is not centered within the specifications, Pp alone may be misleading.

The higher the Pp value…the smaller the spread of the systemâs output. Pp is a measure of spread only. A process with a narrow spread (a high Pp) may not meet customer needs if it is not centered within the specifications.

If the system is centered on its target value…Pp should be used in conjunction with Ppk to account for both spread and centering. Pp and Ppk will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Ppk is an index of process performance which tells how well a system is meeting specifications. Ppk calculations use actual sigma (sigma of the individuals), and shows how the system is actually running when compared to the specifications. This index also takes into account how well the process is centered within the specification limits.

If Ppk is 1.0……the system is producing 99.73% of its output within specifications. The larger the Ppk, the less the variation between process output and specifications. If Ppk is between 0 and 1.0……not all process output meets specifications. If the system is centered on its target value……Ppk should be used in conjunction with the Pp index. If the system is centered on its target value, Ppk and Pp will be equal. If they are not equal, the smaller the difference between these indices, the more centered the process is.

The Pr performance ratio is used to summarize the actual spread of the system compared to the spread of the specification limits (upper and lower). The lower the Pr value, the smaller the output spread. Pr does not consider process centering.When the Pr value is multiplied by 100, the result shows the percent of the specifications that are being used by the variation in the process. Pr is calculated using the actual sigma (sigma of the individuals) and is the reciprocal of Pp. In other words, Pr = 1/Pp.

The Cp index is used to summarize a system's ability to meet two-sided specification limits (upper and lower). Like Cpk, it uses estimated sigma and, therefore, shows the system's potential to meet the specifications. However, it ignores the process average and focuses on the spread. If the system is not centered within the specifications, Cp alone may be misleading. The higher the Cp value…..the smaller the spread of the systemâs output. Cp is a measure of spread only. A process with a narrow spread (a high Cp) may not meet customer needs if it is not centered within the specifications.

If the system is centered on its target value…Cp should be used in conjunction with Cpk to account for both spread and centering. Cp and Cpk will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Cpk is a capability index that tells how well as system can meet specification limits. Cpk calculations use estimated sigma and, therefore, shows the system's "potential" to meet specifications. Since it takes the location of the process average into account, the process does not need to be centered on the target value for this index to be useful.

If Cpk is 1.0…the system is producing 99.73% of its output within specifications. The larger the Cpk, the less variation you will find between the process output and specifications.

If Cpk is between 0 and 1.0…not all process output meets specifications. If the system is centered on its target value…Cpk should be used in conjunction with the Cp index. Cpk and Cp will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

The Cpm index indicates how well the system can produce within specifications. Its calculation is similar to Cp, except that sigma is calculated using the target value instead of the mean. The larger the Cpm, the more likely the process will produce output that meets specifications and the target value.CrThe Cr capability ratio is used to summarize the estimated spread of the system compared to the spread of the specification limits (upper and lower). The lower the Cr value, the smaller the output spread. Cr does not consider process centering.When the Cr value is multiplied by 100, the result shows the percent of the specifications that are being used by the variation in the process. Cr is calculated using an estimated sigma and is the reciprocal of Cp. In other words, Cr = 1/Cp.

So, which is best to use? Cpk or Ppk? Remember, Cpk is used to determine if the process is capable of meeting specifications in the long term. The process has to be in statistical control. A process that is in statistical control is consistent and predictable. You know what your process will produce (within the range of natural variation) in the future. So, Cpk is a prediction of what your process can do. It is the better of the two indices for process capability. You use Ppk only when you can't determine statistical control or if you are looking at a very short production run. If your process is in control, Cpk and Ppk will be essentially the same.

If a process is stable and predictable, and has consistently provided high Ppk numbers for a long period, the process is a candidate for removing final inspection, which is likely to produce more defects than it finds. The process should then have the input variables controlled, and undergo periodic audits.

Perhaps the most common Capability Study error is requesting suppliers to provide just Cpk, along with shipped goods. Cpk can easily be manipulated by selectively changing the order of the data. To get spectacularly inflated values, simply sort the data. If you are to rely on a single index, then Ppk is the appropriate choice. Authoritative texts allow the substitution of Cpk for Ppk if the process is stable and predictable. This is allowed because in that case, they are equal.

What happens if the process is not approximately normally distributed?

What you can do with non-normal data The indices that we considered thus far are based on normality of the process distribution. This poses a problem when the process distribution is not normal. Without going into the specifics, you can transform the data so that they become approximately normal. A popular transformation is the Box-Cox transformation. Another solution is to use or develop another set of indices, that apply to nonnormal distributions.

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