# DefinitionCp Cpk Pp and Ppk Definitions - I'm doing SPC and Capability Analysis

L

#### Leigh

Hi All,

I am fairly new to the world of Capability Analysis. My problem at the moment is the more I read the more my head spins. I’m slowly getting a hold on the concepts involved but I need help on a specific issue.

A customer of ours manufactures an electronic component for us. This component (an IC or integrated circuit) must meet over 150 parameters within well defined limits. (Each lot consists of about 2500 pieces). Therefore the supplier tests the IC prior to shipping and provides, as part of the test report, the capability analysis results for each parameter. The test processes are quite stable and the measured test results are generally centered within the limits. Below is his description of Cp and Cpk. To my understanding he is using actual standard deviation rather than estimated std as is required by Cp /Cpk formulas, and therefore I think are really descriptions of Pk and Ppk..

My question is: 1) Is his interpretation of Cp and Cpk below correct and relevant? 2) Is below actually a description for Pk and Ppk? and 3) Should he be using Cp or Pp in determining out process capability, and if so how?

Many thanks for any help offered…. Leigh

Definition of Cp
===========

Cp, the capability of a process, is determined by the width of the process distribution
relative to a set of limits. For a test program, this means the width of a distribution of
test results relative to the test limits. The standard measure of the width of a
distribution in statistics is the standard deviation (std_dev), so Cp must somehow
relate the standard deviation to the test limits:

Cp = min{ ((HTL-nominal)/(3*std_dev)) , ((nominal-LTL)/(3*std_dev)) }

with:
HTL = Higher Test Limit
LTL = Lower Test Limit
nominal = nominal value = (HTL+LTL)/2
std_dev = standard deviation of measured value

As the equation suggests, Cp is the number of standard deviations that will fit between
the upper and lower limits of a test. A standard convention is to scale the number of
standard deviations by multiplying by 6. This makes it easier to interpret the Cp value
in terms of 6 sigma quality goals. The correct interpretation of Cp is "the number of 6
sigma distribution which can be fit between the high test limit and low test limit".

In actually calculating Cp, Test Insight uses the distance between the nominal value
and the closest test limit. Because this distance covers only half the distribution, it
divides the distance by 3 sigma’s, not six. The end results, however, correspond to the
number of 6 sigma distributions that can fit between the limits.

Definition of Cpk
============

This measures the shift in the mean of a test away from its ideal position. If Cp is a
measure of the potential yield of a perfectly centered distribution, then we should
diminish Cp for every unit the mean deviates from the center of the test limits. It is
this combination of Cp and k which yields the Cpk statistic.

Cpk = Cp (1 - k)

with:
k = (nominal-mean) / (HTL-LTL)/2
mean = mean of measured value

A

#### Al Dyer

Cp = Rating of how much of the specification limits are being used.

USL-LSL / 6 Est. Sigma (Rbard2)

Cpk = Rating of how close your process is to the expected/targeted value.

The lower of:

USL - Xbar / 3 Est. Sigma (Rbard2)

or

Xbar - LSL / 3 Est. Sigma (Rbard2)

The higher the CP, the less of the spec. limits being used. The higher the Cpk the closer you are to the target. For Pp and Ppk you would just replace the Rbard2 sigma with the sigma of the population (sigma hat). I'm not a pro so I'm sure some of the stat people can put in in better statistical terms.

[This message has been edited by Al Dyer (edited 30 March 2001).]

B

#### Big Red

What I was always taught was that Ppk is based on your actual standard deviation whereas Cpk is based on d2 or estimated standard deviation. That is why you use Ppk for short run studies and Cpk for long run studies. Most of our customers want us to submit Ppk for sample submissions only, then Cpk for regular production.

A

#### Al Dyer

I guess I don't differentiate between the two all that much, meaning they need to be used together to get a better picture of the process. (We do use Ppk during submission and then move to Cpk when process is stable)

If I generate the same set of data and see a substantially (don't ask me to define substantially) higher Cpk than a Ppk it leads me to think that my subgroups are not picking up the "fliers" in the population therefore I have not identified/removed all of the special causes of variation in the process. At this point I would go back to the FMEA for further review.

ASD...

L

#### Leigh

Hi Guys...

Thanks for your responses, it has helped my understanding of this hard to grasp subject. And thanks for a great forum!!

Regards - Leigh

C

#### Coleman Donnelly

Hi All,

I am fairly new to the world of Capability Analysis. My problem at the moment is the more I read the more my head spins. I’m slowly getting a hold on the concepts involved but I need help on a specific issue.

A customer of ours manufactures an electronic component for us. This component (an IC or integrated circuit) must meet over 150 parameters within well defined limits. (Each lot consists of about 2500 pieces). Therefore the supplier tests the IC prior to shipping and provides, as part of the test report, the capability analysis results for each parameter. The test processes are quite stable and the measured test results are generally centered within the limits. Below is his description of Cp and Cpk. To my understanding he is using actual standard deviation rather than estimated std as is required by Cp /Cpk formulas, and therefore I think are really descriptions of Pk and Ppk..

My question is: 1) Is his interpretation of Cp and Cpk below correct and relevant? 2) Is below actually a description for Pk and Ppk? and 3) Should he be using Cp or Pp in determining out process capability, and if so how?

Many thanks for any help offered…. Leigh

Definition of Cp
===========

Cp, the capability of a process, is determined by the width of the process distribution
relative to a set of limits. For a test program, this means the width of a distribution of
test results relative to the test limits. The standard measure of the width of a
distribution in statistics is the standard deviation (std_dev), so Cp must somehow
relate the standard deviation to the test limits:

Cp = min{ ((HTL-nominal)/(3*std_dev)) , ((nominal-LTL)/(3*std_dev)) }

with:
HTL = Higher Test Limit
LTL = Lower Test Limit
nominal = nominal value = (HTL+LTL)/2
std_dev = standard deviation of measured value

As the equation suggests, Cp is the number of standard deviations that will fit between
the upper and lower limits of a test.
A standard convention is to scale the number of
standard deviations by multiplying by 6. This makes it easier to interpret the Cp value
in terms of 6 sigma quality goals. The correct interpretation of Cp is "the number of 6
sigma distribution which can be fit between the high test limit and low test limit".

In actually calculating Cp, Test Insight uses the distance between the nominal value
and the closest test limit. Because this distance covers only half the distribution, it
divides the distance by 3 sigma’s, not six. The end results, however, correspond to the
number of 6 sigma distributions that can fit between the limits.

Definition of Cpk
============

This measures the shift in the mean of a test away from its ideal position. If Cp is a
measure of the potential yield of a perfectly centered distribution, then we should
diminish Cp for every unit the mean deviates from the center of the test limits. It is
this combination of Cp and k which yields the Cpk statistic.

Cpk = Cp (1 - k)

with:
k = (nominal-mean) / (HTL-LTL)/2
mean = mean of measured value
Just so i understand this correctily - does this mean that as Cpk of 1 will mean 1 standard deviation of all of my product will fall within tollerance, and that approx. 40% of my product will fall out of my tollerance limits?

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