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SPC (Statistical Process Control) for Unilateral Tolerance


Involved In Discussions
I dont know what is the procedure for conducting spc trial for unilateral tolerance. Can any one explain what is the procedure with example pl.

David DeLong

Re: SPC for Unilateral tolerance

The only difference is that one could only calculate a Ppk. This is the difference between the process average and the specification limit divided by 3 estimated standard deviations. That's about it.

One would have the same sample size -minimum 30 parts, etc.


Forum Moderator
Staff member
Re: SPC for Unilateral tolerance

I dont know what is the procedure for conducting spc trial for unilateral tolerance. Can any one explain what is the procedure with example pl.
Are you referring to SPC as in control charts or to a capability study?

Unilateral tolerances are frequently but not always associated with non-normal distributions. This is sometimes due to the existence of a natural limit or boundary that cannot be crossed such as zero.

Therefore, you must determine whether the data is normal or non-normal and proceed from there. Normal data may use standard charting methods for SPC, while non-normal data may require special methods such as transforming the data. The central limit theorom will tend to normalize the averages and may allow use of Xbar and R charts, but the presence of a boundary could result in impossible limits such as a negative lower limit when it is impossible to pass zero.

If your data is normal, you may use the appropriate Cpk or Ppk measure, but if your data is non-normal, these indices will be incorrect.


Quite Involved in Discussions
Well..., what's "trial"?
IMHO SPC can be done the same for uni as for bilateral tolerances.

as for capability indicators, the lack of one spec, does the calculation not good, not only because of the normality but as Cpk or ppk take in account the existence (or centering on specs) of both. I would prefere Cpmk because it takes in account the target and is a better indicator (IMHO).:bonk:


Quite Involved in Discussions
Dear Folks

This is the article I found on net. may be useful.

Calculating capability indices with one specification

The following formula for Cpk is easily found in most statistics books.
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:
A. Not calculating Cpk since you don’t have all of the variables.
B. Entering in an arbitrary specification.
C. 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.

If you have questions or comments regarding calculations of capability indices, call technical support at 800-777-5060 or send e-mail to [email protected]


Steve Prevette

Deming Disciple
Staff member
Super Moderator
Don't confuse SPC (the voice of the process as Dr. Wheeler puts it) with specifications (the voice of the customer, Wheeler).

We may have a single-sided specification, but it is still worth understanding the variation in the process. For example, I may have a specification to respond within 30 minutes. There is still a difference between a response time that varies between 1 minute and 29 minutes unpredictably and a response time that varies from 25 to 29 minutes. In the second case, I can make the statement to the customer that we will provide a response between 25 and 30 minutes from now, allowing the customer to go off and do something else in the next 25 minutes.

Jim Wynne

Staff member

would you explain in details

I'm not speaking for Steve, of course, but calculating a capability index number is done to compare the specification limits with the statistical control limits in order to make a prediction about how well the process will meet the specifications. The specifications are the "voice of the customer."

SPC per se is done to evaluate the statistical stability of the process, regardless of specification limits, thus whether the specification is unilateral or bilateral is irrelevant. You can chart values and look at their variation in relationship to central tendency without knowing anything about the spec limits. This is the "voice of the process" that Steve (and Wheeler) refer to.

The goal is a happy marriage of the two voices--a process that is predictably able to meet the customer's requirements. Make the process stable, compare it to the customer requirements, then adjust as necessary.
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Stop X-bar/R Madness!!
You can certainly use SPC to track the "voice of the process" with a control chart. Typically, I find most unilateral tolerances most closely behave as a Weibull distribution. That is handy when calculating capability. The worse off you are, the more it behaves like a normal distribution and "regular" SPC rules can apply.
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