Process capability without variability in data?

[Dannyboy175]

My company uses Cpk to judge process capability, with a 1.33 Cpk being the acceptable minimum for most processes. In some cases, we do not have enough variation in the data to calculate SD and Cpk. The issue is often that method/tool used to make a measurement does not provide enough discrimination based on the actual variation, and the SD cannot be calculated. For example, we measure the ID of tubing with a pin gage and all of the readings are the same.

My question is whether there is an accepted alternate judge of process capability that doesn't use standard deviation? Are there any other statistical solutions? I realize that changing the measurement tool/method to get more variation would work, but may not be practical in some situations.

Thanks for the help. This is my first post/question...

Dan

[BrenH]

I'll answer this from a physics perspective, rather than statistical, so be wary, and take it with a big grain of salt.

The measurement taken must have an error, which would be + or - half the resolution limit of the instrument. Since you cannot tell any values below that resolution limit, why not assign random values to it, i.e add the result with a random (below resolution) error component. That way the SD will be able to be calculated, but would probably give values around the magnitude of resolution limit. As it is, you don't really know if you are doing that, measurement-wise, in the first place.

To the statistical experts here, is there anything wrong with that approach?

[Dannyboy175]

I just don't know if there is an accepted scientific/statistical basis for this. Is this kosher?

I'd like to know if anyone else has this problem and how they deal with it.

Of course, if we had pin gages that measured to a greater resolution, then we might capture the variation. Maybe we just have to acquire better equipment.

[BrenH]

Reviewing the equation has led me to a more satisfying result -

Cpk = (USL - LSL)/(6 * SD) - |m - Xbar|/(3 * SD)

where USL is the upper specification limit
LSL is the lower specification limit
m is the nominal value of specification
Xbar is the process average, and
SD is the standard deviation

therefore, if every measurement is the same, Cpk will be infinity. This is fine, so long as you aren't violating the assumptions of the model.

Practically, if USL or LSL are closer to m than the resolution of the measurement instrument, then either the upper or lower limits are not appropriate or the measuring instrument cannot perform to the specifications required.

Now, if the above criterion is satisfied, what else can I know? Since the purpose of Cpk is to identify when a process is falling short, then it is the worst case scenario that we are looking for. In other words, I'm asking the question, if the measurement system that is giving the exact same result every time is both accurate and creating the worst possible SD for the measurement resolution, what would the SD be? The answer to this the same as giving every second measurement an additional 1/2 the resolution limit, and every other measurement have 1/2 the resolution limit subtracted. Then the Cpk can be no worse than the value determined.

I'll do an example. The resolution of an ID tube measurement is +- 0.5 mm. The measurements are 6, 6, 6, 6, 6, 6, 6, 6. Then the worst SD possible for these measurements is the SD for the following set, 6.5, 5.5, 6.5, 5.5, 6.5, 5.5, 6.5, 5.5.

So this method then gives a lower limit for Cpk, which is probably what you want.

Now this probably should be done for all measurement systems, not just ones that give very close results. For a general system, the worst case SD is obtained when 1/2 resolution is added to all measurements above the mean and 1/2 resolution is subtracted to all measurements below the mean, while measurements equal to the mean are alternately added and subtracted 1/2 resolution. This then captures both the variability of the process, and the potential maximum (hidden) variability of the measurement system.

[Dannyboy175]

Your suggestion makes a lot of sense. I'll have to see if its going to hold water in my organization. It would be better received if coming from a published source, as opposed to an internet forum, I think. I appreciate your help, nonetheless.

[Dannyboy175]

Calculating normal Cpk is not difficult. Our problem, however, also includes data with some, but limited variation, where the distribution is not normal. The statistics package we use, Statistica, has multiple non-normal distributions which it tries to fit to. When the data does not have enough variation, then it can't match a distribution and can't calculate Cpk.

I'm going to try to get some data, both normal and nonnormal, and use your half resolution suggestion with it and see what happens.