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Calculating capability indices with one specification

The following formula for Cpk is easily found in most statistics books, as well as in software products such as SQCpack.

Cpk = Zmin / 3

Zmin = smaller of Zupper or Zlower

Zupper = [(USL – Mean) / Estimated sigma*]

Zlower = [(Mean – LSL) / Estimated sigma*]

Estimated sigma = average range / d2

And, we’ve all learned that generally speaking, the higher the Cpk, the better the product or process that you are measuring. That is, as the process improves, Cpk climbs.

What is not apparent, however, is how to calculate Cpk when you have only one specification or tolerance. For example, how do you calculate Cpk when you have an upper tolerance and no lower tolerance?

When faced with a missing specification, you could consider:

  1. Not calculating Cpk since you don’t have all of the variables. Entering in an arbitrary specification.
  2. Ignoring the missing specification and calculating Cpk on the only Z value.

An example may help to illustrate the outcome of each option. Let’s assume you are making plastic pellets and your customer has specified that the pellets should have a low amount of moisture. The lower the moisture content, the better. No more than .5 is allowed. If the product has too much moisture, it will cause manufacturing problems. The process is in statistical control.

It is not likely your customer would be happy if you went with option A and decided not to calculate a Cpk.

Going with option B, you might argue that the lower specification limit (LSL) is 0 since it is impossible to have a moisture level below 0. So with USL at .5 and LSL at 0, Cpk is calculated as follows:

If USL = .5, X-bar = .0025, and estimated sigma = .15, then:

Zupper = [(.5 – .0025) / .15] = 3.316,

Zlower = [(.0025 – 0) / .15] = .01667 and

Zmin = .01667

Cpk = .01667 /3 = .005

Your customer will probably not be happy with a Cpk of .005 and this number is not representative of the process.

Example C assumes that the lower specification is missing. Since you do not have a LSL, Zlower is missing or non-existent. Zmin therefore becomes Zupper and Cpk is Zupper/3.

Zupper = 3.316 (from above)

Cpk = 3.316 / 3 = 1.10.

A Cpk of 1.10 is more realistic than .005 for the data given in this example and is representative of the process.

As this example illustrates, setting the lower specification equal to 0 results in a lower Cpk. In fact, as the process improves (moisture content decreases) the Cpk will decrease. When the process improves, Cpk should increase. Therefore, when you only have one specification, you should enter only that specification, and treat the other specification as missing.

An interesting debate (well, about as interesting as statistics gets) occurs with what to do with Cp (or Pp). Most textbooks show Cp as the difference between both specifications (USL – LSL) divided by 6 sigma. Because only one specification exists, some suggest that Cp can not be calculated. Another suggestion is to look at ~ ½ of the Cp. For example, instead of evaluating [(USL – Mean) + (Mean – LSL)] / 6*sigma, instead think of Cp as (USL – Mean) / 3*sigma or (Mean – LSL) / 3*sigma. You might note that when you only have one specification, this becomes the same formula as Cpk.

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Is Cpk the best capability index?

Cpk has been a popular capability index for many years and perhaps because of its momentum it continues to remain popular. But is it the best index to use? Answering this question assumes that there is one best index, which is a different discussion altogether. Let’s agree that there several other useful capability indices. Two other indices that can be beneficial are Ppk and Cpm. As mentioned in a previous article on Cpk, “Cpk or Ppk: Which should you use,” Cpk uses only the estimated sigma to measure variation. While this is acceptable, the estimated sigma can be artificially low depending on the subgroup size, sample interval, or sampling plan. This in turn can lead to an over-inflated Cpk. For a process that drifts, such as the process shown in the chart below, the estimated sigma will usually be artifcially low. This is because the estimated sigma looks at only variation within subgroups.

Ppk, on the other hand, uses the standard deviation from all of the data. We can call this the sigma of the individual values or sigmai. Sigma of the individual values looks at variation within and between subgroups. For a process that exhibits drifting, estimated sigma would not pick up the total variation in the process and thus the Cpk becomes a cloudy statistic. In other words, one can not be sure it is a valid statistic. In contrast to Cpk, Ppk, which uses the sigma of the individual values, would pick up all the variation in the process. Again, sigmai uses between and within subgroup variation. So if there is drifting in the process, sigmai would typically be larger than the estimated sigma, sigmae, and thus Ppk would, as it should, be lower than Cpk.Here is a quick review of the formulae for Cpk and Ppk:

Cpk = Zmin/3 where Zmin
Ppk = Zmin/3 where Zmin
Zmin = (USL – Mean) / est.sigma
Zmin = (USL – Mean) / sigmai
or = (Mean – LSL) / est. sigma
= (Mean – LSL) / sigmai

 

We should be concerned with how well the process is behaving, therefore Ppk might be preferred over Cpk. Ppk is a more conservative approach to answering the question, “How good is my process?” Watch for a future article discussing the relatively new capability index, Cpm, and how it stacks up against Cpk and Ppk.

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Capability analysis: Indices

What are the measures of process capability?

These indices show what the process is “capable” of doing. This is the six sigma  range of the inherent variation of a statistically stable process. Sigma is estimated using the /d2 formula.

What are the measures of process performance?

These indices show what a process is actually doing. This is the six sigma range of the total process variation, where sigma is usually determined by the sample standard deviation.

All of these capability indices can be calculated using software packages like SQCpack.

See also:
>> Can a process produce output within specifications?
>> Control vs capability
>> Normal data capability analysis
>> Non-normal data capability analysis
>> What is capability analysis and when is it used?
>> Learning more about capability

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How do we determine process capability if the process isn’t normal?

Cp, Cpk, Pp, and Ppk do not necessarily rely on the normal distribution. If any of these indices increases, you know that the process capability has improved. What you do not know is how that improvement translates into good product. This requires knowledge of the distribution of the individual units produced by the process. The Central Limit Theorem refers to averages and this works for the control chart, but it doesn’t work for the histogram. Therefore, we generally make the assumption of the normal distribution in order to estimate the percent out of specification (above, below, and total).

For non-normal distributions, we first estimate some parameters using the data. We then use these parameters and follow a Pearson curve fitting procedure to select an appropriate distribution. Since the relationship between the standard deviation and the percent within CAN vary differently from the normal distribution for distributions that are not normal, (e.g., plus and minus one sigma may not equal 68.26%, plus and minus 2 sigma may not equal 95.44%, etc.), we try to transform the capability indices into something comparable. With this distribution equation, we integrate in from the tails to the upper and lower specifications respectively. Once the percent out-of-spec above and below the respective spec limits are estimated, the z values (for the normal with the same mean and standard deviation) associated with those same percents are determined. Then, Cpk and Ppk are calculated using their respective estimates for the standard deviation. This makes these values more comparable to those that people are used to seeing.

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capability analysis references

Consult the following references:

  • Practical Tools for Continuous Improvement, Volume 1 Statistical Tools, PQ Systems, Inc. Copyright 2000. 1-800-777-3020
  • Fundamental Statistical Process Control, AIAG/ASQC Automotive Industry Action Group, American Society for Quality Control, Copyright 1991.
  • TQT Improvement ToolsTotal Quality Transformation®, PQ Systems, Inc. Copyright 1992. 1-800-777-3020
  • TQT Total Quality ToolsTotal Quality Transformation, PQ Systems, Inc. Copyright 1992. 1-800-777-3020
  • Beyond Capability Confusion, Donald J. Wheeler, Ph.D. SPC Press, Copyright 1999.

See also:
>> Can a process produce output within specifications?
>> Capability vs control
>> Normal data capability analysis
>> Non-normal data capability analysis
>> What is capability analysis and when is it used?
>> What are the capability indices?

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Capability analysis for normal data

The calculations for capability analysis are based on the following assumptions:

  1. The data is normally distributed. In other words, the shape shown by the histogram  follows the “normal” bell curve.
  2. The system being studied is stable and no assignable causes for variation are present. A control chart of the system has been made to determine stability before a capability analysis is done.
  3. The mean of the system being studied falls between the upper and lower specification limits defined for the process.

If these assumptions are not met, the results of a capability analysis will be misleading.

See also:
>> Can a process produce output within specifications?
>> Capability vs control
>> Normal data capability analysis
>> Non-normal data capability analysis
>> What is capability analysis and when is it used?
>> What are the capability indices?

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Capability vs. control

A process is said to be in control or stable, if it is in statistical control. A process is in statistical control when all special causes of variation have been removed and only common cause variation remains.

Control charts are used to determine whether a process is in statistical control or not. If there are no points beyond the control limits, no trends up, down, above, or below the centerline, and no patterns, the process is said to be in statistical control.

Capability is the ability of the process to produce output that meets specifications. A process is said to be capable if nearly 100% of the output from the process is within the specifications. A process can be in control, yet fail to meet specification requirements. In this situation, you would need to take steps to improve or redesign the process.

See also:
>> Can a process produce output within specifications?
>> Capability vs control
>> Normal data capability analysis
>> Non-normal data capability analysis
>> What is capability analysis and when is it used?
>> What are the capability indices?

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Capability analysis for non-normal data

Since a capability study makes the assumption that the data being analyzed is normally distributed, what can be done if the data is not normally distributed?

Usually if the data is not normally distributed, the process is not in control and a capability study is premature. However, in some cases the non-normal process is due to a measure that legitimately has only a single-sided specification. For example, if you are measuring flatness, the measurements can never be smaller than 0. In these cases, you will need to use Pearson curve fitting. Pearson curve fitting is a technique in which the distribution is compared to one of many theoretical distributions. If the data matches closely enough, it will pass a chi-square test and the capability indices will be useful. As with normally distributed data, if the data does not match one of the theoretical distributions, then the capability indices may be misleading and should not be used.

See also:
>> Can a process produce output within specifications?
>> Capability vs control
>> Normal data capability analysis
>> Non-normal data capability analysis
>> What is capability analysis and when is it used?
>> What are the capability indices?

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When should you recalculate limits?

Eventually, everyone using SPC charts will have to decide whether they should change the control limits or leave them alone. There are no hard and fast rules, but here are some thoughts to help you make your decision.

The purpose of any control chart is to help you understand your process well enough to take the right action. This degree of understanding is only possible when the control limits appropriately reflect the expected behavior of the process. When the control limits no longer represent the expected behavior, you have lost your ability to take the right action. Merely recalculating the control limits, however, is no guarantee that the new limits will properly reflect the expected behavior of the process either.

  1. Have you seen the process change significantly, i.e., is there an assignable cause present?
  2. Do you understand the cause for the change in the process?
  3. Do you have reason to believe that the cause will remain in the process?
  4. Have you observed the changed process long enough to determine if newly-calculated limits will appropriately reflect the behavior of the process?

You should ideally be able to answer yes to all of these questions before recalculating control limits.

To create control charts and easily recalculate control limits, try software products like SQCpack.