What process variation component do Cp, Cpk capture?

Cp and Cpk account for within subgroup variation only. This can be seen from the formulae used for the calculations…





In the above formulae process sigma is calculated using the formula given below:


where
Rbar is the average range of the total subgroups
d2 is a statistical constant based on the subgroup size
R or Range is the difference between maximum and minimum of the readings within a subgroup i.e. it accounts for the variation within subgroup.

Thus Cp, Cpk account for within subgroup variation and do not consider subgroup drift in the data.
To find capability of processes considering within subgroup variation as well as subgroup drift, use the Pp, Ppk indices.

How do you measure process capability? What indices do you use?

Measure Cpk only for stable processes

Cpk and other capability indices should be calculated for stable processes only. This is a critical requirement for calculating capability indices. Sadly, it is the most ignored fact in the real world.

What is a stable process?
A process that does not have any special causes present is a stable process. Such a process has inherent variation due to common causes only. The control chart for a stable process has no 'Out of control' points.

Why is stability important for Capability studies?
The capability indices are designed to determine how capable an ongoing process is for consistently delivering product within customer specifications given the natural variation of the process.
Ongoing processes should typically have only common cause variation i.e. the process is statistically stable.
When assignable causes come into play, the process becomes unstable. Now the control chart has out of control points that need to be watched closely and removed as soon as possible.

When processes are unstable, it is recommended to use performance indices like Pp and Ppk.

For more information on capability indices, please read our White Paper Use of different capability indices.

Cp and Cpk are related. Really?

Process capability analysis plays an important role in quality. It helps determine whether the process is capable of consistently producing parts conforming to customer requirements. Cp and Cpk are the two most popular capability indices used to find process capability. Cp and Cpk are related to each other. Whenever I say this most people around me are surprised, some think I am insane. But it is true. Cp and Cpk are related by the equation. Refer to chapter 6 in the NIST/SEMATECH e-handbook of Statistics Handbook

Cpk = (1-k) Cp
where k is the distance between the process mean and the midpoint of specification range.



where
m = midpoint of the specification range or tolerance
mu = process mean
USL and LSL – upper and lower specification limits.

k has a value between 0 and 1.
When k is 0, Cpk is equal to Cp.
When k is less than 1, Cpk < Cp

That is why Cpk ≤ Cp.

Can Cp be used for unilateral tolerances?

When I go through the topic of capability indices for unilateral tolerances in my training class or consulting sessions, I usually get surprised looks from quality practitioners searching for Cp values. The common reaction is where is the Cp index? How come it doesn’t show on the chart or on my capability report?

For unilateral tolerance, Cp does not apply.
Why?
Because, by definition Cp is estimates process capability for a centered process. For unilateral tolerances the target may not be at the center of the specification limits.
The formula for calculating Cp is USL - LSL / 6σ
For unilateral tolerances, we have either the upper or lower specification limit.
That is why Cp cannot be calculated.

So what capability indices can be used for unilateral tolerances?
Cpk, Cpkm can be used. For details on these indices, please download our white paper on capability indices.
Cpk is Cpu for upper specified unilateral tolerance and it is Cpl for lower specified.

How do you track capability of your processes?

Does sample size affect Cpk?

This was a question from one of my training class participants.
I found the question interesting and answered using the formula of Cpk.

Cpk = Minimum of [USL - Overall Average]/3sigma or [Overall Average - LSL]/3sigma
For Xbar-R chart,
Sigma = Rbar/d2
d2 increases as sample size increases.
So for the same Rbar, sigma reduces as sample size increases.

Cpk increases as sigma reduces.
That means for the same overall process average and range, Cpk will change as sample size changes. In fact as sample size increases, Cpk will be larger.

Example

Outside diameter of a pipe has specifications of 9 and 11. Design target is 10

For a dataset, let's say that...
Overall average = 9.3 and Rbar = 0.5

We will now calculate Sigma and Cpk for two different sample sizes - 5 and 7

For sample size 5, d2 = 2.326
sigma = 0.5/2.326 = 0.215
Cpk = Min [ 10 - 9.3] / 3 x 0.215 or [9.3 - 9] / 3 x 0.215
Cpk = Min of 1.09 or 0.465
Cpk = 0.465

For sample size 7, d2 = 2.704
sigma = 0.5/2.704 = 0.185
Cpk = Min [ 10 - 9.3] / 3 x 0.185 or [ 9.3 - 9] / 3 x 0.185
Cpk = Min of 1.26 or 0.54
Cpk = 0.54

For the same data set, sigma and Cpk change based on the sample size selected. As sample size increases, sigma reduces and Cpk increases.
If a smart quality engineer wants to trick the system and project good process capability, he/she can simply do that by increasing the sample size.

What is the sample size you use? Have you tried finding process capability with different sample size?
Drop us a line at info@sybeq.com and tell us about your process and capability calculations. We would love to hear from you.

A picture is worth thousand words...

I always thought that pictures were more effective and efficient in communicating a message. People around the world come up with more and more innovative ways to present business data as pictures that are easy to read and convey the message quickly.

In the Quality world, I deal with pictures in the form of charts(SPC charts, affinity diagrams, relation diagrams etc) or maps (SIPOC maps, VSM etc). These pictures (charts, diagrams and maps) capture lot of meaningful information about the
quality of the product or service. However, many quality practitioners struggle with proper interpretations of these pictures and fail to take necessary action to improve quality.

Control charts for example are designed to communicate what's happening in the process. They provide signals that indicate the presence of special causes in the process. All we have to do is pick up these signals and act on them!

Rather than looking through paper reams full of data that could take hours, pictures tell us the real story in just a few minutes.
A picture is truly worth a thousand words...but will be effective only if it is looked at by the right people, interpreted correctly and acted upon.

Do quality charts or maps help you find quality problems in your process?
Have you improved a process by acting on signals from SPC charts?

Is my data variable or attribute?

From my training class last week, I had an interesting question from one of my class students who was from the healthcare industry.

The student asked "I measure success of tests conducted by lab personnel. For example, I monitor number of blood tests were conducted every day by each lab assistant. I also monitor how many attempts were required for the assistant to successfully draw blood and complete the test.
I want to plot an SPC chart for the number of successful blood draws in single attempt. Should I use variable or attribute control charts?"

My answer to her was a series of questions..
Question 1: What do want the SPC chart to tell you?
Answer : Number of successful blood draws in single attempt.
Also, success rate of each lab personnel.

Question 2: Are either of the above measurements or counts?
Answer : They are counts.

Question 3: What SPC chart helps chart count data?
Answer: Attribute control charts

My student was able to answer her own question after breaking down her problem in small sections.

Have you struggled with data types?
Do you have questions about type of SPC charts you should use?
Please visit our tutorial on "SPC in healthcare" at www.sybeq.com/Tutorial.aspx