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In the Six Sigma quality methodology, process performance is reported to the organization as a sigma level. The higher the sigma level, the better the process is performing.
Another way to report process capability and process performance is through the statistical measurements of C_{p}, C_{pk}, P_{p}, and P_{pk}. This article will present definitions, interpretations and calculations for C_{pk }and P_{pk} though the use of forum quotations. Thanks to everyone below that helped contributed to this excellent reference.
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Definitions
C_{p}= Process Capability. A simple and straightforward indicator of process capability.
C_{pk}= Process Capability Index. Adjustment of C_{p} for the effect of noncentered distribution.
P_{p}= Process Performance. A simple and straightforward indicator of process performance.
P_{pk}= Process Performance Index. Adjustment of P_{p} for the effect of noncentered distribution.
Interpreting C_{p}, C_{pk}
“C_{pk} is an index (a simple number) which measures how close a process is running to its specification limits, relative to the natural variability of the process. The larger the index, the less likely it is that any item will be outside the specs.” Neil Polhemus
“If you hunt our shoot targets with bow, darts, or gun try this analogy. If your shots are falling in the same spot forming a good group this is a high C_{p}, and when the sighting is adjusted so this tight group of shots is landing on the bullseye, you now have a high C_{pk}.” Tommy
“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.” Ajit
“You must have a C_{pk} of 1.33 [4 sigma] or higher to satisfy most customers.” Joe Perito
“Consider a car and a garage. The garage defines the specification limits; the car defines the output of the process. If the car is only a little bit smaller than the garage, you had better park it right in the middle of the garage (center of the specification) if you want to get all of the car in the garage. If the car is wider than the garage, it does not matter if you have it centered; it will not fit. If the car is a lot smaller than the garage (Six Sigma process), it doesn’t matter if you park it exactly in the middle; it will fit and you have plenty of room on either side. 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.” Ben
“The value itself can be thought of as the amount the process (car) can widen before hitting the nearest spec limit (garage door edge).
C_{pk }=1/2 means you’ve crunched nearest the door edge (ouch!)
C_{pk }=1 means you’re just touching the nearest edge
C_{pk }=2 means your width can grow 2 times before touching
C_{pk }=3 means your width can grow 3 times before touching” Larry Seibel
Interpreting Pp, Ppk
“Process Performance Index basically tries to verify if the sample that you have generated from the process is capable to meet Customer CTQs (requirements). It differs from Process Capability in that Process Performance only applies to a specific batch of material. Samples from the batch may need to be quite large to be representative of the variation in the batch. Process Performance is only used when process control cannot be evaluated. An example of this is for a short preproduction run. Process Performance generally uses sample sigma in its calculation; Process capability uses the process sigma value determined from either the Moving Range, Range or Sigma control charts.” Praneet
Differences Between C_{pk} and P_{pk}
“C_{pk} is for short term, P_{pk} is for long term.” Sundeep Singh
“P_{pk} produces an index number (like 1.33) for the process variation. C_{pk} references the variation to your specification limits. If you just want to know how much variation the process exhibits, a P_{pk} measurement is fine. If you want to know how that variation will affect the ability of your process to meet customer requirements (CTQ’s), you should use C_{pk}.” Michael Whaley
“It could be argued that the use of P_{pk} and C_{pk} (with sufficient sample size) are far more valid estimates of long and short term capability of processes since the 1.5 sigma shift has a shaky statistical foundation.” Eoin
“C_{pk} tells you what the process is CAPABLE of doing in future, assuming it remains in a state of statistical control. P_{pk} tells you how the process has performed in the past. You cannot use it predict the future, like with C_{pk}, because the process is not in a state of control. The values for C_{pk} and P_{pk} will converge to almost the same value when the process is in statistical control. that is because sigma and the sample standard deviation will be identical (at least as can be distinguished by an Ftest). When out of control, the values will be distinctly different, perhaps by a very wide margin.” Jim Parnella
“C_{p} and C_{pk} are for computing the index with respect to the subgrouping of your data (different shifts, machines, operators, etc.), while P_{p} and P_{pk} are for the whole process (no subgrouping). For both P_{pk} and C_{pk} the ‘k’ stands for ‘centralizing facteur’ – it assumes the index takes into consideration the fact that your data is maybe not centered (and hence, your index shall be smaller). It is more realistic to use P_{p} and P_{pk} than C_{p} or C_{pk} as the process variation cannot be tempered with by inappropriate subgrouping. However, C_{p} and C_{pk} can be very useful in order to know if, under the best conditions, the process is capable of fitting into the specs or not.It basically gives you the best case scenario for the existing process.” Chantal
“C_{p} should always be greater than 2.0 for a good process which is under statistical control. For a good process under statistical control, C_{pk} should be greater than 1.5.” Ranganadha Kumar
“As for P_{pk}/C_{pk}, they mean one or the other and you will find people confusing the definitions and you WILL find books defining them versa and vice versa. You will have to ask the definition the person is using that you are talking to.” Joe Perito
“I just finished up a meeting with a vendor and we had a nice discussion of C_{pk} vs. P_{pk}. We had the definitions exactly reversed between us. The outcome was to standardize on definitions and move forward from there. My suggestion to others is that each company have a procedure or document (we do not), which has the definitions of C_{pk} and P_{pk} in it. This provides everyone a standard to refer to for WHEN we forget or get confused.” John Adamo
“The Six Sigma community standardized on definitions of C_{p}, C_{pk}, P_{p}, and P_{pk} from AIAG SPC manual page 80. You can get the manual for about $7.” Gary
Calculating C_{pk} and P_{pk}
“P_{p} = (USL – LSL)/6*Std.dev
C_{pl} = (Mean – LSL)/3*Std.dev
C_{pu} = (USL – Mean)/3*Std.dev
C_{pk}= Min (C_{pl}, C_{pu})” Ranganadha Kumar
“C_{pk} is calculated using an estimate of the standard deviation calculated using Rbar/d2. P_{pk} uses the usual form of the standard deviation ie the root of the variance or the square root of the sum of squares divided by n – 1. The Rbar/D2 estimation of the standard deviation has a smoothing effect and the C_{pk} statistic is less sensitive to points which are further away from the mean than is P_{pk}.” Eoin
“C_{pk} is calculated using RBar/d2 or SBar/c4 for Sigma in the denominator of you equation. This calculation for Sigma REQUIRES the process to be in a state of statistical control. If not in control, your calculation of Sigma (and hence Cpk) is useless – it is only valid when incontrol.” Jim Parnella
“You can have a ‘good’ C_{pk} yet still have data outside the specification, and the process needs to be in control before evaluating C_{pk}.” Matt


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Comments
Excellent summary!
I am better understanding cp &cpk
Excellent explanation of process measuring methodology. It explains clearly each and every stage. Good.
K Kumar
Cp should always be greater than 2.0 for a good process which is under statistical control. For a good process under statistical control, Cpk should be greater than 1.5.
If below 2.0 for Cp and Cpk 1.5 may i know why ?
If over 2.0 for Cp and Cpk 1.5 may i know why ?
Very useful article on CP And CPK.
Have better understanding after reading this article, Thanks.
Very Useful Article.. It helps me a lot in understanding the meaning and concept of Process Capability…Now I can start explaining this to all..Thanks a lot
I asked the below question of a software provider hoping he could explain why his software was giving me a Cpk greater then 1.0 while having points out of spec? Can any one out there help me better understand ?
“…I still do not understand how Cpk can be greater than one with a point out of spec. your template must be wrong. There is no way that you can have a CPK (not CP) greater than one with point (s) out of spec???????
His reply: “Your question is incorrect”
“Cp and Cpk do not take into account where the process mean is located relative to the specifications. It simply measures the spread.”
Montgomery Intro to SPC 4th.
Cp and Cpk answer the question “should it fit”, not “does it fit.”
If your “outofspec” point is a “flyer” meaning that the balance of your population is much greater (or less) than this flyer, then the calculation will still give you a Cpk greater than 1. However, rule #1 is your process must be “under control” before you can caluculate a Cpk. Therefore it may indeed be that your process is not under control and in this case all bets are off.
Cpk measures process centering based on mean and STDEV not range. Depending on the n (number of samples) in your data set a single outlier will have a large or small affect. Remember the formula is this Cpk = (Mean LSL or USL if lower number is generated)/3*Std. The comments the formula doens’t work for outliers is partialy correct. Calculation of a Cpk is a performance metric to show process control. It doesn’t incorporate western electric rules. Also Cpk1.33 means 64 defects per million and your process is running at 4 sigma. Link to conversion table provided.
http://www.amplastics.com/cpk_vs_ppm.htm
Victor
Technically, your software provider is right (although not very helpful). The Cp & Cpk calculation is based on the process mean & range and has nothing to do with how many points are in or out of spec. In actual practice, this shouldn’t happen very often. The Cpk is an indicator of how centered your process is (use Cp and Cpk together to evaluate this).
The Cpk calculation assumes that the data is normally distributed. From the sounds of it, your data may not be. A skewed distribution would throw off the range. If you haven’t checked for normality, I’d start there.
i have a big doubt, in calculation of cpk does it matter if the process acts like nonormal process or normal process, wich curve should i take weibull, longest side, etc then if my process is unilateral how do i need to calculate the procees performance???
very good explaination..
the information given is very useful
very good explanation
BECAUSE IF A PROCESS IS UNDER CONTROL IT DOES NOT ALWAYS MEAN THAT IT IS CAPABLE.
What is difference between Standard Deviation and Estimate Standard Deviation. Why Estimate Standard Deviationcalculated in SPC.?
how can you calculate Pp or Ppk when your standard deviation is 0?
You typically wouldn’t calculate Pp or Ppk when st. dev. is 0. Pp/Ppk would be infinitely large. Either your process is so good, any actual variation is small as compared to the specification or your inspection/measurment method is invalid. Ensure measurement system analysis rules are followed.
What is difference between Standard Deviation and Estimate Standard Deviation. Why Estimate Standard Deviationcalculated in SPC.?
I think this answers your question:
Standard Deviation = Population and Estimate Standard Deviation = Sample
Estimate St Dev is used in SPC because that’s what you are calculating for, a sample of the actual population.
Hello all,
I am new to this forum. I have some question regarding capability metrices.
USL – 1.35
LSL – 1.11
Observation Results – 1.15, 1.15, 1.16, 1.15, 1.15, 1.16, 1.15, 1.15, 1.16
Mean – 1.1533
Std Dev – 0.005
Cp = (1.35 – 1.11/ 6 * 0.005)
= 8
CpK = Min (1.35 – 1.1533/ 3 * 0.005, 1.1533 – 1.11/ 3 * 0.005)
= Min (13.11, 2.88)
CpK = 2.88
After reading materials for capability index, it was clear that value above 2 for Cp and CpK is great. Process has achieved 6 sigma capability.
But as we see my results are towards Lower Specification Limit. All are skewed to left of the plot.
Does still my process is known as 6 sigma capable?
Or
My calculation is wrong?
Please clarify.
Thanks,
Bharat
You are presenting a very small sample size. I would first use this formula to calculate the needed n values, but if your data holds yes you do have a six sigma process. THe first formula you used tells you if it were centered you have a 8 sigma possibility, but perform lower because you are off center.
http://www.isixsigma.com/toolstemplates/samplingdata/howdeterminesamplesizedeterminingsamplesize/
Bharat Change your USL and LSL
I have a process for Incident Management ,Which handles different Customers and different Customers has different SLA like 240Mins .300Mins,480Mins ,720Mins etc…..
Can anyone please help me in calculating the Cp and Cpk of this Process.
LSL=0
USL=? i am confused what is the USL bcoz it has different SLA’s.
Plese help me in resolving this its urgent……..
The USL and LSL are the spec requirements that You put on the process. i.e. Above or below the USL or LSL then the product/performance is not fit for use. UCL and LCL are determined statistically depending upon how tight you wish/need to control the process.
bharat,
think ur sd calculation is wrong.
Ano,
SD calculation is right.
I did it in Excel (0.005) as well as using math formula (0.0047).
Bharat
Bharat,
0.005 is variance. SD is square root of variance=0.070711
and cpk =min(0.92,0.204) = 0.204
Ano,
Sorry for late response.
There is some confusion…
Let me show my calculation for standard deviation. I know Std.Deviation is Sq Rt of Variance.
Observation Results – 1.15, 1.15, 1.16, 1.15, 1.15, 1.16, 1.15, 1.15, 1.16
Mean – 1.1533
Std Dev = Sq.Rt [ (1.15  1.1533)^2 + (1.15  1.1533)^2 + (1.16  1.1533)^2 + (1.15  1.1533)^2 + (1.15  1.1533)^2 + (1.16  1.1533)^2 + (1.15  1.1533)^2 + (1.15  1.1533)^2 + (1.16  1.1533)^2/ 9]
6 observations of 1.15 and 3 observations of 1.16
(1.15 – 1.1533)^2 = 0.00001089 X 6 = 0.00006534
(1.16 – 1.1533)^2 = 0.00004489 X 3 = 0.00013467
Total = 0.0002000.1/9
= 0.0000222
Std Deviation = Sq rt (0.0000222)
Std Deviation = 0.0047
Bharat, i think your calculations are right, i worked out the same using minitab and excel as well..
TOTAL / MEAN 9.23 1.15
MIN / MAX 1.15 1.16
STD. DEV 0.0052
USL / LSL 1.35 1.11
Cp 7.73
CpU / CpL 12.64 2.82
CpK 2.82
Regards,
Ravi
please note that while calculating the formula for standard deviation do not use N. Use N1. Here i see you divided your value by 9. Sample size become 8. you can use minitab for the correct results. however, use hands until you are familiar
Bharat
I agree, I think your calculations are correct
I think the problem is your small sample size.
If i do a minitab calculation of
Power and Sample Size
Test for Two Standard Deviations
Testing (StDev 1 / StDev 2) = 1 (versus not =)
Calculating power for (StDev 1 / StDev 2) = ratio
Alpha = 0.1
Method: Levene’s Test
Sample
Size Power Ratio
9 0.9 4.55704
9 0.9 0.21944
The sample size is for each group.
This means that with only 9 samples, your 90% confidence limit for your standard deviation is a ratio between
0.2 and 4.5
I.E your actual standard deviation could be between
0.001 and 0.0225
worst case (0.0225), your CP values come out at
Min(2.91, 0.64)
If it was my process, I would take more samples
e.g
Sample
Size Power Ratio
50 0.9 1.59061
50 0.9 0.62869
I would also try to increase the precision of your measurements, as you could expect the difference between 1.15 and 1.16 to be measurement error….
So in short, the issue probably lies in your SD figure, not in your calculations but your method.
The analogy we used at Motorola in the mideighties when Six Sigma was devloped:
Imagine we are the road crew painting a series of white stipes that form dotted lines down the center of the highway.
With just a sampling of as little as three stripes, CP calculates the consistency of one stripe to another, meaning – “How exactly alike are they?”
And CPK measures how centered we are making the sequence of stripes that constitutes our dotted line – “Is it exactly up the center?” In the manufacturing business this would mean “How centered is each piece between its + and – tolerances.
CP – The more the slight variations, meaning the looser our process control becomes (people fatigue in the hot sun, or because the machine occasionally sputters and splatters) the more likely the stripe will soon become unacceptably long, short, or wide, thereby creating a statistical number of rejects per thousand, or per million, etc. And…
CPK – The less centered we are, (as our vision starts to blur, or the truck wobbles to one side) the sooner we can expect to reduce the lane size so that it is too small for cars to fit, or maybe we will even run off the road!
Hope this helps explain it to nonQA’ers.
Really good description of Cpk & Ppk
I highly recommend all of you to quit using software. You need to understand the fundementals of this to even being to know what any answer means. All mathematical calculations should be performed by hand only using a calculator. By doing this you will understand what the answer means.
Sir,
Very useful informations.
Thanks and regards.
Suresh Rawal
Very apt articulation of the subject matter – the concept has been explained quite lucidly. Thanks a ton
good conclusion to understand cp/cpk fundamental.
To sum it up…
There are NO definitions for Cp, Cpk, Pp and Ppk stated that is valid everywhere and accepted by all.
What is NORMAL definitions in Automotive Industry is whats in AIAG handbooks.
Short version of this is
Pp = (USL LSL)/6*Std.dev
Cp = (USL LSL)/6*Std.dev
Ppl/Cpl = (Mean LSL)/3*Std.dev
Ppu/Cpu = (USL Mean)/3*Std.dev
Ppk/Cpk= Min (Cpl, Cpu)
Difference between Cp and Pp indexes are HOW TO CALCULATE THE SIGMA.
Sub groups over time = Cp indexes / All data available = Pp
Special case for Cp; NO sub groups (individual data points/sub group size equal to “1”) BUT not “long term” evaluation. In this case You use table approximations of the estimations of SIGMA.
Then it is up to You to define what is long term, short term and if to use subgroups or individual data points.
Criterias for these descisions are time, destructive/nondestructive testing for the data, price of parts etc.
So if You collect data one Monday morning, say one sample every 15 min for 4 hours You will get 13 samples.
If the parts are expensive You might just collect one measurement in the case of destructive testing or just one since the measuring is time/cost consuming. In this case You will get an estimate of the SIGMA within the “sub group” by assuming process in control and stable and using table values (this is done automatically by most software such as MINITAB with sample size set to 1). The OVERALL Sigma is calculated as any normal SD.
If You have sampled say 5 data point each time then You can calculate a WITIN subgroup Sigma (datapoint vs sub group mean for each sub group).
So now we have an estimated WITHIN subgroup SIGMA and an OVERALL SIGMA.
Sigma within (based on table values or sub group calculations) is used for Cp and Cpk calculations.
Sigma overall is used for the Pp and Ppk calculations.
The same goes for if You would have collected the data for a whole week, say one sample on Monday one on Tuesday and so on. Same calculations but different analysis.
So, what is “long term” and what is “short term” has to be defined.
And logically the meaning of the four different capability indexes are;
Ppk is what the customer sees.
Pp is what Ppk would have been if the process were centered.
Cpk is what the Ppk will be if the process is stable (no change between subgroups mean).
Cp is what Ppk will be if the process is stable AND centered.
Sorry to tell, but definition of Cpk and Ppk is wrong, it´s just the other way round:
Ppk is a short term analysis (mostly done with 50 or 100 samples measured)
Ppk is Process Potential Capability
Cpk is long term analysis and is Continous Process capability
i.g. Automotive Industrie requires a Ppk better than 1,67 and Cpk better than 1,33
or Ppk better than 2,0 and Cpk better than 1,67 (f.e. daimlerfor safety criterias)
its good and easy to understand.
Very good summary
Very good summary
Very Useful Article.It helps me a lot in understanding the meaning and concept of Process Capability.Thanks a lot
Very Useful Article.It helps me a lot in understanding the meaning and concept of Process Capability.Thanks a lot