Studying the Capability of Capability Studies - Part 1

Ok. I admit I was trying to be funny in my post title. But the issue in today's blog is "how capable is the capability study?" Or to state another way, "when should we be careful in how much we trust our capability study results?"

Here are three items we should be aware of when designing, running and calculating a Capability study.

• Cp versus Pp
• The distribution of the data
• Sample size

This post will discuss Cp versus Pp. Future post will cover distribution of the data and Sample size issues.

First lets agree to just discuss overall capability, i.e. Tolerance Range / 6*sigma. Everything discussed below will apply to Cpk and Ppk as well.

Cp versus Pp

Okay. Diving right in. As many of you already know, the difference between Cp and Pp is in  how the standard deviation (sigma) is calculated.

Cp uses a standard deviation calculated from the "Within" variation (when there are logical sub-groups).

Pp uses all the data regardless of sub-groups to calculate the standard deviation. 

As an example, assume I am printing solder paste onto a PCB (for the non-electronics people, think of silk screen printing of T-shirts, only not really).

I could do one of three things. 

1. Select one pad and measure the paste thickness, then repeat that same measurement for each sequential PCB until I reach my desired sample size. 
1. Cp (using 'Within' sigma) would be calculated from the Moving Range (ala I-mR charts). 
2. Pp would be calculated using the total data set and the standard formula for sigma.
2. Select several pads on the PCB (maybe five total, one on each side and one in the middle) and measure thickness of all these pads for each PCB used in the capability study.  
1. Cp would use "within" sigma calculated from the pooled standard deviation of the five pads in each PCB across every board. (Pooling is similar to averaging the standard deviations of each sub-group. Its complicated. Another post for the future.
2. Pp would, as before, use the entire data set and the standard formula for sigma.
3.  Collect measurements of paste thickness for each PCB in the Capability study but do not keep track of the order of the build, or sub-groups of any kind. 
1. Cp is not possible in this case. Warning! Minitab will still assume the data is in time order and calculate a Cp but in this case Cp should be ignored. 
2. Pp - yep. Use all the data.

The "within" variation (sigma) will always be equal to or smaller that the sigma calculated from the entire data set.  Therefore Cp will always be equal or higher than Pp. Or, one can think that the Cp will contain only the common cause variation and is therefore one measure of how good the process can become if all the special causes are eliminated. Pp contains both common cause and special cause variation and is one measure of how the process is doing today. 

It may help to see an example (below) to understand the difference. Let pretend we are measuring drilled holes in a PCB.

The Minitab  charts below show "drilled hole" data that is normal/ We also see from the Time series plot that there is a gentle descent of the data over time -- a special cause event which could be due to drill wear. 

We can also see that the Cp =  36.42 while Pp = 1.45. 

We can interpret these results to mean that as of today our process has a capability of 1.45, but if we could identify and eliminate the special cause condition we could be as good as 36.42. Maybe not the best example, but I hope it help illustrate the concept. 

Quite often if you -- as a customer -- asks for capability data, you will get a list of data and a "Cp" value. You need to always question the supplier if the data is in time order or if there are logical sub-groups. Is the capability reported actually Pp?

Bottom line is that many people speak of and report Cp (or Cpk) values when they are really giving us Pp (or Ppk).  The differences are subtle, but can be critical.

Use the comments to give me your thoughts.