Cpk vs Ppk

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    Pradeep Sharma

    Cpk vs Ppk
    As per my understanding, when we measure process capablity we consider Cpk for normal data and Ppk for non normal data, and when we consider Ppk, we also look at parts per million….however parts per million is derived from normal distribution, so why do we consider Ppk for non normal data when it is based on normal distribution? What is the statistical difference between Cpk and Ppk?



    Your understanding is incorrect.  Follow the link below and do a little research.  Good luck.


    Pradeep Sharma

    Thanks for the my next question is how do we calculate process capability if we have non normal data.


    Robert Butler

    For non-normal take your data and plot it on normal probability paper and identify the .135 and 99.865 percentile values (Z = +-3).  The difference between these two values is the span for producing the middle 99.73% of the process output.  This is the equivalent 6 sigma spread and the capability goal of this spread is to have this equal to .75 x Tolerance.  If you have software that permits you to fit Johnson distributions it will find these values for you but if you don’t the above will permit you to do it by hand.
      For more details  try Measuring Process Capability by Bothe.  Chapter 8 is titled “Measuring Capability for Non-Normal Variable Data.


    The Force

    For non normal data, using minitab, you can see te process capability using a weibull distribution



    *Normal data Cap Studies gives you Cpk & Ppk.
    *NonNormal data Cap Studies gives only Ppk.
    * Ppk  = Preliminary Process Cap
    * Cpk  = Process Cap

    Ppk = an average of 1 population /  Long-term performance-data.
    Ppk, on the other hand, uses the standard deviation from all of the data. We can call this the sigma of the individual values or sigmai. Sigma of the individual values looks at variation within and between subgroups.
    Ppk  > 1.67
    Ppk is an index of process performance which tells how well a system is meeting specifications. Ppk calculations use actual sigma (sigma of the individuals), and shows how the system is actually running when compared to the specifications. This index also takes into account how well the process is centered within the specification limits.
    If Ppk is 1.0… …the system is producing 99.73% of its output within specifications. The larger the Ppk, the less the variation between process output and specifications.
    If Ppk is between 0 and 1.0……not all process output meets specifications
    When data is Non-Normal use Weibull and Pp.
    * Cpk = an average of averages / several population averages /  Short Term Performance-data / Normal Data

    Cpk = Z(short-term) which is sigma level / 3
    Cp= Spec Range / Process Cap  USL-LSL/6XSTD
    Cpk uses only the estimated sigma to measure variation.
    Cpk  > 1.33
    Cpk is a capability index that tells how well as system can meet specification limits. Cpk calculations use estimated sigma and, therefore, shows the system’s “potential” to meet specifications. Since it takes the location of the process average into account, the process does not need to be centered on the target value for this index to be useful.
    If Cpk is 1.0……the system is producing 99.73% of its output within specifications. The larger the Cpk, the less variation you will find between the process output and specifications.
    If Cpk is between 0 and 1.0……not all process output meets specifications
    Cpk means “production capacity”, and Ppk means “production performance”
    Estimated sigma and the related capability indices (Cp, Cpk, and Cr) are used to measure the potential capability of a system to meet customer needs. Use it when you want to analyze a system’s aptitude to perform.
    Actual or calculated sigma (sigma of the individuals) and the related indices (Pp, Ppk, and Pr) are used to measure the performance of a system to meet customer needs. Use it when you want to measure a system’s actual process performance.
    Cpk=1/2 means you’ve crunched nearest the limits (ouch!) Cpk=1 means you’re just touching the nearest limits Cpk=2 means your width can grow 2 times before touching Cpk=3 means your width can grow 3 times before touching
    CCpk is a potential capability index. It is identical to the Cpk index, but is centered at the process Target (Capability Analysis > Options > Target), if the target is specified. The Cpk index is centered at the midpoint of the specification limits if both the specification limits are given.
         CCpk is precisely Cpk when the target and the one of the specification limits are not given, or if the midpoint of the specification limits and the target are the same.
    DefinitionsCp = Process Capability. A simple and straightforward indicator of process capability.Cpk = Process Capability Index. Adjustment of Cp for the effect of non-centered distribution.Pp = Process Performance. A simple and straightforward indicator of process performance.Ppk = Process Performance Index. Adjustment of Pp for the effect of non-centered distribution.
    *Capabilities Study (yes/no) = only apply to Customer Specifications.
    Most Engineers consider a capable process to be one that has a Cpk of 1.33 or better, and a process operating between 1.0 and 1.33 is “marginal.” Many companies now suggest that their suppliers maintain even higher levels of Cpk. A Cpk exactly equal to 1.0 would imply that the process variation exactly meets 3 Sigma. A Cpk exactly equal to 1.33 would imply that the process variation exactly meets 4 Sigma. If the process shifted slightly, and the out of control condition was not immediately, if not sooner, detected, then the process would produce scrap. This is the reason for the extra .33. It allows for some small process shifts to occur that could go undetected.
    Process Capability and Defect Rate :
    Using process capability indices it is easy to forget how much of product is falling beyond specification. The conversion curve presented here can be a useful tool for interpreting Cpk with its corresponding defect levels. The defect levels or parts per million non-conforming were computed for different Cpk values using the Z scores and the percentage area under the standard normal curve using normal deviate tables.


    Paul White

    I am sorry but I dont agree with the last thread.
    The pre-requisites for calculating both Cpk & Ppk are:

    Stable process (free from special cause)
    Normally distributed
    Both Cpk & Ppk and based on the assumption of normality.
    Which indice to apply always provokes debate. However, I usually use the results of the stability assessment to determine which indice is appropriate.
    The reason for this is due to the way the standard deviation is calculated:

    Cpk – SD calculated by rbar/d2. R bar is taken from the control chart, the within subgroup average range. D2 is Shewharts (grandfather of statistical quality control!) mathmatical constant based on the subgroup sample size.
    Ppk – SD is calculated using all the individual data points but using besel’s correction factor (n-1). N-1 is used because we are dealing with sample data. I.e. We have never captured the full population (see Excel for both methods of calculating SD).
    Therefore, both methods use an estimate of standard deviation. Cpk SD is based on the within subgroup variation from the control chart & Ppk SD is based on the all the individual data points.
    Thus, if the process is stable (& normal) I will use Cpk as this is method is based on the control chart and will provide an accurate measure of capability.
    If the process is unstable (but normal) I will use Ppk as this method looks at the individual data points and will not be susceptible to special cause and the order of the data.
    However, if the process is non-normal I will first ask why the data is non-normal. If this is to be expected I conduct a box-cox transformation of the data and specification. This will (hopefully) convert the data into a normal distribution to allow a comparison against the converted spec limits. It is always important to test the normality of the converted data before calcultaing Cpk/Ppk.

    Stable & Normal = Cpk
    Unstable &  but normal = Ppk
    Non-Normal = Transformation of data & spec.
    I would be grateful for any comments on this. Am I following the correct approach?


    Robert Butler

      As I said in my first post to this subject – if the system is stable and the data is non-normal then you determine the equivalent six standard deviation spread (either short term or long term) using the method I outlined which is straight from the chapter in the Bothe book.  You don’t need to transform.
      Almost any data from any process with a bound due to a physical limit will be non-normal. – Taper, flatness, surface finish, concentricity, eccentricity, perpendicularity, angularity, roundness, warpage, straightness, etc. all produce in control data which is non-normal.


    Pradeep Sharma

    I would like to share my data with you guys which I plotted on minitab:
    As suggested i used wiebull distribution to calculate process capability as my data was non-normal. I got following Pp and Ppk (my LSL is 0 and ULS is 2880 mins)
    Pp: 0.10
    Ppk: -0.29
    PPU: -0.29
    PPL: 2.40
    what all infrence can I make out from the above result. Please let me know if any body needs more infiormation.
    Robert: I will appreciate if you could elaborate little bit more on the method you suggested. Do I need to calculate Z values for my data and get the Sigma level using Z-value, if yes then why use PPk to represent process performance, we can present that by Sigma level. Please help



    Your data says that your average is above your USL and you have way to much variation.
    If your specifications are rational, stop worrying about how to report a capability metric and go figure out why your average and standard deviation are so high.



    You may want to test if your data fits a Weilbull distribution first. Minitab 14 allows you to test if the Weillbull is appropriate.
    however, first question is why is it non-normal? Skewed? Outliers etc?
    Also, try converting the data & specification using a box – cox transformation in Minitab. What is he Pp & Ppk now?
    But, as Stan points out pp close to zero means your spread of data is almost as wide as the spec and negative ppk means your mean is located outside  the USL. The Box cox will allow you to see this graphically (but the actual values will be different)

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