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:
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.
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:
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:
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: