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

*C*_{pk} tells us what a process is capable of doing in future, assuming it remains in a state of statistical control.

*P*_{pk} 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;

The values for *C*_{pk} and *P*_{pk} will converge to almost the same value because sigma and the sample standard deviation will be identical (use an F test to determine).

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

*C*

_{p}value. When the you have a tight group of shots is landing on the bulls eye, you now have a high

*C*

_{pk}

*C*

_{pk}measures how close you are to your target and how consistent you are to around your average performance. A person may be performing with minimum variation, but he can be away from his target towards one of the specification limit, which indicates lower

*C*

_{pk}, whereas

*C*

_{p}will be high. On the other hand, a person may be on average exactly at the target, but the variation in performance is high (but still lower than the tolerance band (i.e., specification interval). In such case also

*C*

_{pk}will be lower, but

*C*

_{p}will be high.

*C*

_{pk}will be higher only when you r meeting the target consistently with minimum variation

## 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. *C*_{pk} 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:

*C*_{pl} = (Process Mean – LSL)/(3*Standard Deviation)

*C*_{pu} = (USL – Process Mean)/(3*Standard Deviation)

Cpk is merely the smallest value of the Cpl or Cpu denoted: *C*_{pk}= Min (*C*_{pl}, *C*_{pu})

### 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

*C*_{pk}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
*C*_{pk}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?

*C*_{pk }= Negative number: Your process will regularly crash the car into the wall.

*C*_{pk }=0.5: You have a good chance hitting the wall on entry.

*C*_{pk }=1: Your car may be just touching the nearest edge of the entry.

*C*_{pk }=2: Great! You have great clearance. You could double the width of your car before you hit the side of the garage.

*C*_{pk }=3: Excellent! You have excellent clearance. You could triple the width of your car before you hit the side of the garage.

### Cpk Video

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.

### Capability Index

How do Cp, Z values, DPMO , Specification Limits, Standard Deviation, and Capability all relate?

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:

(A) –0.25

(B) 1.00

(C) 1.33

(D) 1.67

**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.

A confidence interval is a statistical measure used in hypothesis testing and is not pertinent to this question.

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.

Joanna Han says

Hi, if I have a set of data where the subgroup size is different, how should I determine which d2 value to be used for the Cpk calculation? If I perform a Ppk calculation, is the Ppk value going to be affected by the difference in subgroup size? Thanks.

Six Sigma Study Guide says

Hi Joanna – Not sure I’m following your first question. Are you asking which d2 value to choose if you have multiple subgroups of varying size? (Ex. subgroup 1 has 5 elements, 2 has 4, 3 has 5?)

However, Ppk values shouldn’t be affected by subgroup size as you don’t use it in the calculation. – See this article on Ppk calculation.

Joanna Han says

Yes. My first question was about how to determine d2 for multiple subgroups of varying size, i.e. what you have given in the example.

Six Sigma Study Guide says

Joanna, you’ve asked a great question and I’m going to need to study a bit more before I can answer.

If you were designing the sampling, I’d suggest controlling it so that your subgroups were the same size. Since we all know that in practice we often inherit data, so this may not be possible. My instinct would be to take the average of the subgroup sizes. So if we had sizes of 5, 4, 5 – I’d round up and use 5.

I’ll investigate further and see what I find. A friend suggested I check the text Statistical Quality Control by Grant and Leavenworth. Trying to get my hands on a copy now.

Joanna Han says

Hi, I am doing data analysis where the subgroup size is not constant. I was thinking to take the majority (or average) but I have found that there is formula to determine the sigma. Have a look here: http://elsmar.com/pdf_files/Capability%20Analysis%20Formulas%20-%20Minitab.pdf

Praful bhusari says

WHAT IS DIFFERENCE between CP and CPK?

Six Sigma Study Guide says

The Shooting at a Target Analogy

A good analogy is shooting at a target. If the rounds form a good cluster or grouping in the same spot anywhere on the target you have a high Cp value. When the you have a tight group of shots is landing on the bulls eye, you now have a high Cpk

Cpk measures how close you are to your target and how consistent you are to around your average performance. A person may be performing with minimum variation, but he can be away from his target towards one of the specification limit, which indicates lower Cpk, whereas Cp will be high. On the other hand, a person may be on average exactly at the target, but the variation in performance is high (but still lower than the tolerance band (i.e., specification interval). In such case also Cpk will be lower, but Cp will be high. Cpk will be higher only when you r meeting the target consistently with minimum variation

Pavel says

Hello! How was out of spec percentage (2.28%) calculated in a video lesson?

Six Sigma Study Guide says

Hi Pavel, it was calculated using a Z Score. See some more on the relationship between Process Capability and Z scores here: http://sixsigmastudyguide.com/z-score-and-process-capability/

Then, see how to do the equation by following the examples here: http://sixsigmastudyguide.com/z-scores-z-table-z-transformations/

PARAG PANDYA says

Can we calculate Cp & Cpk for components batch size of 2 or 3.

Six Sigma Study Guide says

Yes, Parag. See the notes that Joanna Han left above.

Abdul Ahad says

process capability studies

Six Sigma Study Guide says

Abdul – not exactly sure what you’re looking for here. How can I help?

bhushan kumar says

cpk value is 1.12 , is it process is good?

Six Sigma Study Guide says

It’s ok. It’s not great.

ddot.re says

best summarized Cp Pp Cpk PPk chart that capsulizes dignificant facotrs

Six Sigma Study Guide says

No idea what you are asking for.

Abhinav singh says

What is the value of Cpk for six sigma process

Six Sigma Study Guide says

It’s an equation, Abhinav. Read the article.

prabin says

A process that has a Cp ≥ 1 and a Cpk ≤ 1, is

Six Sigma Study Guide says

What do you think, Prabin?

karen says

Can you solve the problem?

The weights of nominal 1-kg containers of a concentrated chemical ingredient are shown in Table 8E.2. Suppose there is a lower specification at 0.995 kg. Calculate an appropriate process capability ratio for this material. What percentage of the packages produced by this process is estimated to be below the specification limit?

weights of containers

0,9475 0,9775 0,9965 1,0075 1,018

0,9705 0,986 0,9975 1,01 1,02

0,977 0,996 1,005 1,0175 1,025

Six Sigma Study Guide says

Yes, I can. What have you tried so far?

chandana says

How many number of batches Require for caculation of Cp & CPk Valules in Anual product quality review

Six Sigma Study Guide says

Hi Chandana, What do you think would be appropriate and why?

Gianfranco says

Hi,

I have a doubt about the table under the “Capability Index” paragraph. It links Cp and Z, and there is a constant Cp=Z/3. My question is should not be Cpk=Z/3?, instead for Cp should be Cp=Z/6. Thanks in advance

Six Sigma Study Guide says

Hi Gianfranco, Why should it be Cp = Z/6? Where do you see that?

Amit says

Hello,

I would like to know more whether we can calculate process capability of Manual processes & what are the rules to calculate manual process capability (Theory).

Thank you so much.

Erica says

Hi,

1.How much OK ratio there is in Cp=1.33 and Cpk=1.33?

2.How much OK ratio there is in Cp=1.0 and Cpk=1.0?

Thank you

Six Sigma Study Guide says

Erica, I’m not sure what you mean by OK ratio. Can you elaborate?

Guru says

Hi,

I have a data which contains the quality scores of the individual persons from last 50 weeks (Individual scores for 50 persons on 50 weeks).

Can I use the cpk calculation to know how many persons are in USL & LSL?

Or any other method will be used ?

Please suggest. Thanks in advance.

Six Sigma Study Guide says

Guru, I’m not entirely sure what you are asking. Can you add a bit more detail?

Guru says

Hi,

Sorry for my unclear question before.

Currently, I have been measuring the quality for a group of staffs on a weekly basis. I also set a bandwidth that the staffs who scored more than 90 percentage were good and less than 90% was bad.

Now I have the data history for the last one year. If I want to see the statistical detail for the past one year data(which means can I able to say the sigma levels for each staff) what method will be used?

Six Sigma Study Guide says

Interesting question. Is the sample size homogeneous? Were the same people measured for every test or did the population change over time? Was the test the same each time?

Guru says

Thanks for the comments.

1.The sample size was homogeneous.

2.Mostly same people were measured for every test. Sometimes, the new people were added and will be added overtime.

3.The population will slightly change every time.

4.The testing method will be the same each time.

Six Sigma Study Guide says

Hard for me to give a straight answer without knowing more details on what kind of analysis you will be looking to do, but here are some thoughts:

Since this looks like attribute data (pass / fail), consider treating it like so and forget the scores. Then use an attribute chart to show changes over time where each fail is a “defective” not a “defect”. You’d calculate baseline sigma like so.

Look at what Jeremy did in his case study on using control charts on student test scores. Depending on your use, you might consider an EWMA chart.

If you want to compare the different populations against each other, consider a MANOVA.

Guru says

Hi

Thanks very much for the detailed response.

Now I will start my analysis with the baseline sigma.

However, I will consider the other sources for my future analysis plan

Thanks again

ashley says

How does a Cpk of 2.0 translate into 6 sigma

quality?

• Must show the calculations supporting your response.

Six Sigma Study Guide says

This is a good homework question because it shows the relationship between process capability and quality. Start by listing the Cpk formula and substituting the 2.0 value. What are you left with?

Ashley says

It is a homework question. I put CPK=Z/3=6/3=2 and he said it wasnt good enough. He said, define the Cpk and Z score formulas first.

Then start applying your substitutions. Begin with the given, i.e. Cpk = 2.0.

From there, see how to elegantly interconnect the Z score in the Cpk formula.