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Process Capability with Cp, Cpk, Pp, Ppk and Z Score

Process Capability SixPackZ value, Cp, Cpk, Pp and Ppk can be quite confusing to understand and a task in itself to remember these concepts for Six Sigma Green Belt Project Leads and  other project or process managers who want to understand their current and potential performance.  Lets talk about each of these terms individually.

Z value helps you understand your process yield.  Ideally for a six sigma process, you will have mean separated from upper and lower specification limit by 6 sigma.  So Z value of 6 would indicate your process is performing at 6 sigma level (Z Value is equal to your process current Sigma level).  Pp score of 2 also indicates the same thing.  Z is calculated by using below formula:

Z value using upper specification limit: (Upper Specification Limit-Mean)/Std. Dev.

Z value using Lower specification limit: (Mean – Lower Specification Limit)/Std. Dev.

*Note: we need to calculate Z Value for both ends because the process might not be centered and the Z value using upper and lower specification limit might be different.  This would help you calculate the rejection %age/ defects produced above upper specification limit and below lower specification limit.  Process Capability ExampleThe example output given here indicates different Z values towards USL and LSL.  Using this Z value, you can either refer to Z Table to get the rejection %ages towards right and left side of distribution.  Alternatively, you can calculate the same using Minitab (Calc> Probability Distribution> Normal.  In the window that opens up enter mean of sample and standard deviation.  Under input constant, enter the value of Lower Specification Limit and choose ‘Cumulative Probability’.  The reject percentage will be given in session window.  Please note that using same method you can get rejection percentage for Upper Specification Limit also.  However, the number that you get our of session window needs to be subtracted from 1 to get rejection percentage towards right.).  If you do not have access to minitab, you can download excel template available under six sigma downloads in our website that will help you check the reject percentages.  Add total rejection percentage and subtract it from 1 to get yield details.

Process CapabilityNow lets speak about Cp, Cpk, Pp and Ppk.  The process capability indices Pp and Cp describe how closely a process can operate within its specification limits.  Pp and Ppk talk about your current capability whereas Cp and Cpk gives details about potential capability.  Pp is calculated as Pp= (USL – LSL) / (6*σoverall)

where σ = standard deviation overall for sample
USL = upper specification limit
LSL = lower specification limit

Cp is calculated as Cp= (USL – LSL) / (6*σwithin) (notice the difference is just that of overall and within)

Where σ = standard deviation within subgroups
USL = upper specification limit
LSL = lower specification limit

Cpk 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.  Cp does not talk about centering.  Even if your Cp is 4 or 5, you might have 100% defects because your process is not centered to specification.  Cpk adjusts that and takes centering into account.  Similar is Ppk which is a index of current capability.

Calculating Cpk and Ppk
Pp = (USL – LSL)/6*Std.dev Overall

Ppl= (Mean – LSL)/3*Std.dev Overall

Cp = (USL – LSL)/6*Std.dev Within

Cpu = (USL – Mean)/3*Std.dev Overall

Cpl = (Mean – LSL)/3*Std.dev Within

Cpu = (USL – Mean)/3*Std.dev Within

Cpk= Min (Cpl, Cpu)

Ppk= Min (Ppl, Ppu)

When to Use Pp, Ppk, Cp, and Cpk

When to use Pp Ppk Cp Cpk

Notes :

1. If X-Bar is at target, then Cp = Cpk.

2. Cpk will always be equal to or less than Cp.

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