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Weibull Wobble? Process Capability Analysis with Nonnormal Data

Manufacturers need to make items that meet a customer’s standards, or they’ll soon be out of business. That’s why quality engineers devote a good deal of time to making sure that processes are able to meet those standards. 

The first step is to make sure your process is stable. After all, you can’t predict the performance of an unstable process. But you can predict and improve on a stable process. 

If we know our process is stable, we can use a statistical technique called process capability analysis to see if the process is capable of consistently producing products that meet customer standards. 

Process Capability with Normal Data

But to deliver an accurate picture, capability analysis needs to match the distribution of the process data. Let’s say you’re a tile manufacturer who needs to keep the degree of warping in a ceramic bath tile between 2 and 8 millimeters. (If you're using Minitab Statistical Software, open the sample data set “Tiles.mtw” if you want to play along.) After collecting data about the process, you choose Stat > Quality Tools > Capability Analysis > Normal… and create the following plot:  

Process Capability with Normal DataNormal Process Capability

Hmmmm. See the problem?  These data clearly don’t follow a normal distribution; none of the data points fall under the far left side of the normal curve. The calculations used to conduct this analysis don’t match the distribution of the data, so we can’t trust the results. A basic normality test confirms that the normal distribution does not model the data well. 

So what should we do? One option might be to transform the data to better match the normal distribution. But we also can see if the data fit an alternative distribution, then use Minitab’s built-in tools for performing capability analysis on nonnormal data. 

Let's see how well the data fit some alternative distributions by using Stat > Quality Tools > Individual Distribution Identification... in Minitab. 

Identify Distribution Plot 

In this case, a p-value below .05 on the Goodness-of-Fit test indicates that a distribution is not a good fit for the data.  As shown above, both the plot and the p-value for the Weibull distribution goodness-of-fit test suggest that it is a good fit for this data.  

Analyzing Process Capability with the Weibull Distribution 

Since we know our data follow the Weibull distribution, we’ll select Stat > Quality Tools > Capability Analysis > Nonnormal… in Minitab Statistical Software to get the following dialog box:

 

 Nonormal Process Capability Software

Now we’ll do a capability analysis using a Weibull distribution, Minitab’s default for nonnormal capability analysis. This distribution can take on the characteristics of other types of distributions, making it extremely flexible in fitting different kinds of data. The Weibull distribution is a common alternative to the normal distribution in the case of skewed data.

Let’s see what we get: 

Weibull process capability analysis

That’s a much better fit! 

The Weibull distribution is defined by three parameters: shape, scale, and threshold. The shape parameter refers to the shape of the Weibull curve: 3 approximates a normal curve, while a low value like the 1.69 in the graph above produces a right-skewed curve. A high shape value for shape, like 10, will result in a left-skewed Weibull curve.

Interpreting the Analysis

We interpret the results of a nonnormal capability analysis just as we do an analysis done on data with a normal distribution. Capability is determined by comparing the width of the process variation to the width of the specification. We would like the process spread to be smaller than, and contained within, the specification spread.

That’s clearly not the case with this data. 

The Overall Capability index on the right side of the graph depicts how the process is performing relative to the specification limits. To quickly determine whether the process is capable, compare Ppk with your minimum requirement for the indices. Most quality professionals consider 1.33 to be a minimum requirement for a capable process. A value less than 1 is usually considered unacceptable.

With a Ppk of .25, it seems our tile manufacturer has more work to do to get this process to the point where it will meet customer specifications.  But at least these data offer a clear understanding of how much the process can be improved!

In the meantime, if you’re interested in learning more about capability analysis, check out these resources: 

Assessing Capability video

Capability Statistics, Part 1

Capability Statistics, Part 2

 

Comments

Name: Omar Mora • Tuesday, November 22, 2011

Dear Eston,
This is a great post. Thanks for sharing.
In my humble experience, I have seen how many professionals (engineers, LEAN Six Sigma Black Belts, etc.) tend to omit MSA and when they find data does not fit a normal pattern, they use non-normal or non-parametric capability analysis as their last resort.
I know Minitab does perform non-parametric capability analysis (using a Macro) and that´s great, but, what do you think about the need to emphasize in performing MSA before asumming a non-normal or non-parametric behavior?


Name: Barb • Friday, March 30, 2012

What do you do if your data fits NONE of the distributions including the Johnson and Box Cox transformations?


Name: Eston • Friday, March 30, 2012

Hi Barb, this is an excellent question, but one that's so dependent on individual situations that it's really hard to offer help via a blog. Can I suggest you contact our Tech Support department, either by phone or online: http://www.minitab.com/support/ We offer customers free support from experts who are highly skilled in statistics. They'd be happy to help you figure out what options you might have.


Name: enrique • Tuesday, December 31, 2013

It is great this turn down THE BIG FALLACY , that IN QUALITY CONTROL EVERYTHING IS NORMAL


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