Cp and Cpk are considered short-term potential capability measures for a process. In Six Sigma we want to describe processes quality in terms of sigma because this gives us an easy way to talk about how capable different processes are using a common mathematical framework. In other words, it allows us to compare apple processes to orange processes!
This is a long article, but I thought it was important to keep Cp and Cpk together. Cpk is addressed first, then Cp. There are also crib notes on what the equations mean in a real performance sense, what you should be able to tell about a process depending on Cp and Cpk values and more. If you are not finding what you are looking for, please let me know in the notes below.
What is the Difference between Cp, Cpk and Pp, PPk?
Cp and Cpk are called Process Capability. Pp and Ppk are called Process Performance. In both cases we want to try to verify if the process can meet to meet Customer CTQs (requirements).
Cp, and Cpk are used for Process Capability. Generally you use this when a process is under statistical control. This often happens with a mature process that has been around for a while. Process capability uses the process sigma value determined from either the Moving Range, Range or Sigma control charts
Pp and PPk are used for Process Performance. Generally you use this when a process is too new to determine if it is under statistical control. Ex. there is a short pre-production run or you are piloting a new process. Because there is not a lot of historical data we take large samples from the process to account for variation. Process Performance generally uses sample sigma in its calculation.
In theory Cpk will always be greater than or equal to Ppk. There are anomalies seen when the sample size is small and the data represents a short amount of time where estimating using R will overstate standard deviation and make Cpk smaller than Ppk. It is not real, there can never be less variation in the long term since the long term is using all of the data not just two pieces of data from every subgroup.
Evaluating process capability with Cp & Cpk mirror what is done (and why it is done) when following the Pp & Ppk approach. The main difference is that you use Cp & Cpk after a process has reached stability or statistical control.
Cp vs Cpk
The ‘k’ stands for ‘centralizing factor.’ The index takes into consideration the fact that your data is maybe not centered.
Cpk vs Ppk
Cpk tells us what a process is capable of doing in future, assuming it remains in a state of statistical control.
Ppk tells us how a process has performed in the past and you cannot use it predict the future because the process is not in a state of control.
If a process is in statistical control;
In other words, if Cpk == Ppk, the process is likely in statistical control.
If a process is NOT in statistical control;
Cpk and Ppk values will be distinctly different, perhaps by a very wide margin.
What is the Difference Between Cp and Cpk?
The Shooting at a Target Analogy
What is Cpk?
The Parking a Car in the Garage Analogy
If the car is a lot smaller than the garage, it doesn’t matter if you park it exactly in the middle; it will fit and you have plenty of room on either side. That’s one of the reasons the six sigma philosophy focuses on removing variation in a process.
If you have a process that is in control and with little variation, you should be able to park the car easily within the garage and thus meet customer requirements. Cpk tells you the relationship between the size of the car, the size of the garage and how far away from the middle of the garage you parked the car.”
How to Calculate Cpk
Cp is an abbreviation. There are really two parts; the upper and the lower denoted Cpu and Cpl respectively. Their equations are:
Cpl = (Process Mean – LSL)/(3*Standard Deviation)
Cpu = (USL – Process Mean)/(3*Standard Deviation)
Cpk is merely the smallest value of the Cpl or Cpu denoted: Cpk= Min (Cpl, Cpu)
Why are we dividing by 3 to find Cpk?
Calculating Cpk using a Z Value
If you have a Z value, the equation is very easy;
Cpk can be determined by dividing the Z score by three.
A z score is the same as a standard score; the number of standard deviations above the mean.
Z = x – mean of the population / standard deviation.
Notes and Characteristics of Cpk
Cpk and Centered Processes
Notes on Cpk
- Cpk measures how close a process is performing compared to its specification limits and accounting for the natural variability of the process.
- Larger is better. The larger Cpk is, the less likely it is that any item will be outside the specification limits.
- When Cpk is negative it means that a process will produce output that is outside the customer specification limits.
- When the mean of the process is outside the customer specification limits the value of Cpk will be Negative
- We generally want a Cpk of at least 1.33 [4 sigma] or higher to satisfy most customers.
- Cpk can have an upper and lower value reported.
- If the upper value is 2 and the lower is 1, we say it has been shifted to the left.
- This tells us nothing about if the process is stable or not.
- We must report the lower of the 2 values.
What are Good Values for Cpk?
Remember the Car parking in the garage analogy?
Cpk = Negative number: Your process will regularly crash the car into the wall.
Cpk =0.5: You have a good chance hitting the wall on entry.
Cpk =1: Your car may be just touching the nearest edge of the entry.
Cpk =2: Great! You have great clearance. You could double the width of your car before you hit the side of the garage.
Cpk =3: Excellent! You have excellent clearance. You could triple the width of your car before you hit the side of the garage.
Great, clear, concise video on this subject.
“If you were producing a Cpk equal to 1, than you could expect to produce at least 99.73% good parts.”
How to Calculate Cp
Just as you use Cp & Cpk when a process is stable and Pp & Ppk when a process is new, the way you calculate each are a bit different, too.
Pp = (USL – LSL) / 6* s
In Pp, s is the standard deviation, or the ‘fatness’ or dispersion of the bell curve.
In Cp, we replace s with and estimate of σ we call σr. To do that we leverage the Moving Range concept from a Moving R Bar chart or an XMR Chart. So, σr = [ R Bar / d2]
R Bar comes from the Moving range.
D2 reflects values derived from integrating the area under the normal curve. We often use a table which gives a d2 value based on how many subgroups were in the sample.
Cp does not account for centering.
Cp = (USL – LSL) / ( 6* σr )
Cp = (USL – LSL) / ( 6* R Bar / d2 )
Cp for Process Mean close to USL
If your Process Mean (central tendency) is closer to the USL, use: [ USL – x(bar) ] / [3 * R Bar / d2], where x(bar) is the Process Mean.
Cp for Process Mean close to LSL
If your Process Mean (central tendency) is closer to the LSL, use: [x(bar) – LSL ] / [3 * R Bar / d2], where x(bar) is the Process Mean.
Also see Z values and process capability.
Notes on Cp Values
- If the ratio is greater than one, then the Engineering Tolerance is greater than the Process Spread so the process has the “potential” to be capable (depending on process centering).
- If, however, the Process Spread is greater than the Engineering tolerance, then the process variation will not “fit” within the tolerance and the process will not be capable (even if the process is centered appropriately).
Capability Ratio Cr
The capability ration is the inverse of Cp
Cr = 1/ Cp = ( 6* σr ) / (USL – LSL)
If Cr < 0.75, the process is capable.
If Cr = 0.75 – 1.00, the process is capable with tight control.
If Cr >1, the process is not capable.
Notes on Relating Cp And Cpk
- If Cp == Cpk, then the process is perfectly centered. If perfectly centered, Cp == Cpk.
- Because Cpk accounts for centering (where Cp does not), Cpk can never be larger than Cp.
- Both assume a stable process.
ASQ Six Sigma Black Belt Certification Process Capability Questions:
Question: Data being used in the initial set-up of a process are assumed to have a normal distribution. If the nominal (target) is set at the center of the distribution, and the specification limits are set at ±3s from the center, then the Cpk is equal to:
Answer: 1.00 This answer is by definition.
ASQ Six Sigma Green Belt Certification Process Capability Questions:
Question: When calculating the Cp index, what does the standard deviation represent in the formula Cp = (USL – LSL) / 6σ?
(A) The tolerance interval
(B) The confidence interval for the result
(C) The range of the process
(D) The variance of the index
Answer: (C) The range of the process is represented by 6 sigmas – or standard deviations in the Cp Index equation. Remember, 6 standard deviations account for nearly all eventualities (assuming normal distribution) of the process. 6 sigmas is a good approximation of the range of outcomes.
The USL – LSL (the upper and lower limits) could be representative of the tolerance interval. I’m not familiar with anything called the tolerance interval but the upper and lower limits are set by what is considered acceptable by the client. And those fit.
Variance is a specific term in Six Sigma. You can calculate variance by squaring the standard deviation – a term that does not appear in the Cp equation so we can eliminate this answer.