Process Capability (Cp & Cpk)

Process Capability Cp Cpk example

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 Cpk vs Pp Ppk

Cp Cpk vs 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;

The values for Cpk and Ppk 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?

Cp and Cpk measure how consistent you are to around your average performance.

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

What is Cpk?

The Parking a Car in the Garage Analogy

If you think of the walls of your garage – where you have to fit your car in – they become the customer specification limits. If you go past those limits, you will crash, and the customer will not be happy!
If your process has a lot of variation, that means the process average is all over the place. Not good for parking a car, and not good for any other process. To give your parking process the best chance of success you should work on reducing variation and centering.
If the car is too wide for the garage, nothing you do to center the process will help. You have to change the dispersion of the process (make the car smaller.)

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

Cpk is a measure to show how many standard deviations the specification limits are from the center of the process. On some processes you can do this visually. Others require an equation.
To find Cpk you need to calculate a Z score for the upper specification limit (called Z USL) and a Z score for the lower specification limit (called Z LSL).
Since we are trying to measure how many standard deviations fit between the center line and the specification limit you should not be surprised that the value of those limits, the process mean, and the standard deviation are all components of the Z calculation.

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?

We know that any specification limit has an upper bound and a lower bound. Because you know that 6 sigmas – or 6 standard deviations account for nearly all eventualities on a process (assuming normal distribution) you shouldn’t be surprised to see the “/ 3” because we are looking at only one side of the distribution.

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_popZ = x – mean of the population / standard deviation.

 

Notes and Characteristics of Cpk

Cpk and Centered Processes

If a process is perfectly centered, it has a Cp of 1. That would indicate that mean was 3 standard deviations away from the upper limit and the lower limit.
A perfectly centered process – a process who has a mean exactly in between the 2 specification limits (meaning halfway between the two will have a Cpk of 1. How is this possible? Let’s check the math.
If a process is perfectly centered, then we know that the (USL – Process mean) equals the same thing as the (Process Mean – LSL). Let’s call that A.
Z USL = USL – Process Mean / Standard Deviation. then becomes Z USL = A/ Standard Deviation
and
Z LSL = Process Mean – LSL / Standard Deviation then becomes Z LSL = A / Standard Deviation.
The exact same thing.

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.
Process Capability Cp Cpk example

That was poorly centered!

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.

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.

Let’s revisit Pp

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.

d2 subgroup values

d2 subgroup values

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.

 

Capability Index

Capability Index

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.

28 comments… add one
  • Joanna Han

    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.

    • 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

        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.

        • 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.

  • Praful bhusari

    WHAT IS DIFFERENCE between CP and CPK?

    • 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

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

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

  • process capability studies

  • bhushan kumar

    cpk value is 1.12 , is it process is good?

  • ddot.re

    best summarized Cp Pp Cpk PPk chart that capsulizes dignificant facotrs

  • Abhinav singh

    What is the value of Cpk for six sigma process

  • prabin

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

  • karen

    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

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

  • Gianfranco

    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

  • Amit

    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.

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