Cpk computation for Skewed Distribution

B

bulentson

#1
I would like to ask anyone who has an idea on Cpk computation methods that are robust and can work better on SKEWED distribution. :rolleyes:

Classical method which is min (Cpu, Cpl) tends to be very sensitive to outliers especially if the parameters that we're assessing is one-sided (the higher reading the better) - that is why, even though the minimum of the data is way above the lower spec limit but because of high readings that tend to make the distribution skewed to the right, Cpk which is equal to Cpl (in this case of one-sided LSL) is still below 1 which is clearly misleading. :nope:

Currently i'm looking at the pearson curves method which utilizes percentiles - but i am not so sure of the appropriate percentile value for sample size of 30. :bonk:

Can anyone advice for a much better alternative Cpk computation method for skewed data? :thanx:
 
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Tim Folkerts

Super Moderator
#2
Try a data transformation?

How about transforming the data so that it is closer to normal? If the distribution has a long tail to the right, then perhaps taking the log of the data or the square root may transform the data to something approximately normal. Then do the capability analysis on the transformed data. Or standard Box-Cox transformations may do the trick.

Minitab (and I'm sure other software) has such options built in.

Tim F
 
D

Darius

#3
bulentson said:
Currently i'm looking at the pearson curves method which utilizes percentiles - but i am not so sure of the appropriate percentile value for sample size of 30.
You can compute the estimate percentile value with an interpolation into the cumulative relative frecuency chart.:biglaugh:

bulentson said:
Can anyone advice for a much better alternative Cpk computation method for skewed data?
Try Cmpk, Cpm or Taguchi Loss Function (almost the same), the adventage of Taguchi's Loss Function is that you don't care about specs (with the target exception) and don't framed by the famous 1.33.:mg:
 

Statistical Steven

Statistician
Staff member
Super Moderator
#4
Tim Folkerts said:
How about transforming the data so that it is closer to normal? If the distribution has a long tail to the right, then perhaps taking the log of the data or the square root may transform the data to something approximately normal. Then do the capability analysis on the transformed data. Or standard Box-Cox transformations may do the trick.

Minitab (and I'm sure other software) has such options built in.

Tim F
Just a couple of warnings with this approach.

1. Make sure you transform the specifications.
2. The standard deviation needs to be calculated correctly based on your transformation.
3. The capability indices need to be interpreted differently since the variance structure is not normal.

Having said that, since capability analysis is subjective, transforming data works well in most cases.
 

Jim Wynne

Staff member
Admin
#5
Statistical Steven said:
...capability analysis is subjective...
How so, if what we're trying to do is make a prediction of process performance based on emprical (objective) data and basic probability theory?
 

Tim Folkerts

Super Moderator
#6
Steven,

Good points. It is easy to forget these details when the software has the analysis built in, but any calculations by hand (or more likely by spreadsheet) need to deal correctly with transformations of specs and st. dev.

Tim F
 

Statistical Steven

Statistician
Staff member
Super Moderator
#7
JSW05 said:
How so, if what we're trying to do is make a prediction of process performance based on emprical (objective) data and basic probability theory?
Here is why it is subjective in my mind.

From the same process and quality of output, for one customer your Cpk could be 2.0 and for another 1.0. The only difference is the customers specfiications. The Cpk is not objective because it is based on arbitrary specifications that can be different for different customers.

Second reason it is subjective is that depending on how you collect your data, your estimate of the SD can change, altering the Capability analysis.

If capability analysis had some confidence parameter based on the estimate of SD, I would think it was a little less subjective.

Just my thoughts.
 

Jim Wynne

Staff member
Admin
#8
Statistical Steven said:
Here is why it is subjective in my mind.

From the same process and quality of output, for one customer your Cpk could be 2.0 and for another 1.0. The only difference is the customers specfiications. The Cpk is not objective because it is based on arbitrary specifications that can be different for different customers.

Second reason it is subjective is that depending on how you collect your data, your estimate of the SD can change, altering the Capability analysis.

If capability analysis had some confidence parameter based on the estimate of SD, I would think it was a little less subjective.

Just my thoughts.
I understand what you're saying, but I think the confusion lies in the fact that interpretation of Cpk can be subjective, not Cpk itself. If we assume that data collection, verification of control and the math are done correctly, Cpk is what it is--a way of predicting performance in relationship to specification limits. The fact that specification limits might change isn't a reflection on the objectivity of the analysis, on the contrary, the Cpk principle is verified by testing it against different sets of limits.
 

Miner

Forum Moderator
Staff member
Admin
#9
Minitab can provide a capability analysis based on distribution fitting or using the Johnson Transformation.

This is an oversimplified explanation, but the basic theory is to calculate the area under the fitted curve, select a point equivalent to the same area under a Normal curve from the median to 3 StdDev, and use those numbers to calculate Cpk. Depending on how badly the distribution is skewed, the "3 StdDev" value for the lower half versus the upper half can differ dramatically.

If you try to duplicate this method on your own, please visit the Minitab website for the specific method.
 

Tim Folkerts

Super Moderator
#10
JSW05 said:
I understand what you're saying, but I think the confusion lies in the fact that interpretation of Cpk can be subjective, not Cpk itself. If we assume that data collection, verification of control and the math are done correctly, Cpk is what it is--a way of predicting performance in relationship to specification limits. The fact that specification limits might change isn't a reflection on the objectivity of the analysis, on the contrary, the Cpk principle is verified by testing it against different sets of limits.
The challenge is that any predictions about performance require assumptions about the distribution of the data. When you summarize the data to a single number - Cpk - then you are throwing away all the information about the distribution.

For Cpk=1, the standard answer is that 0.3% would be bad (for a normal distribution). It would be quite possible to have a process with Cpk=1 where all he parts fall within the limits (for example, a triangular distribution). It is also possible that several % would be outside the limits (for example, a 16 cavity injection mold - 15 cavities work nearly perfectly, 1 makes parts just outside the specs).

So a given calculated Cpk could be good, bad, or just OK, depending on the actual situation and the actual data.

Tim F
 
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