# Ppk vs Cpk - A Good, clear explanation and How Mini-Tab Handles Certain Statistics

#### David McGan

##### Inactive Registered Visitor
I'm almost embarrassed to ask the question, but I've not been able to find a good, clear explanation of the real difference between the Ppk and Cpk indices. Ppk, I know, is usually specified for short-term study results, and uses the calculated standard deviation in its determination. Cpk, on the other hand, is used for long-term study results, and uses the estimated standard deviation in its calculation. But can anyone give me the statistical rationale for this?

#### Marc

##### Retired Old Goat
Staff member
I'm embarrassed to answer after 3 years or so, so I guess we're belatedly even...

I was reviewing some posts during a search and found this oldie. Well, most questions get answers (sooner or later)...

I can't give you a statistical rational - maybe Don will see this and comment.

The differences are:

Pp

The Pp index is used to summarize a system's performance in meeting two-sided specification limits (upper and lower). Like Ppk, it uses actual sigma (sigma of the individuals), and shows how the system is actually running when compared to the specifications. However, it ignores the process average and focuses on the spread. If the system is not centered within the specifications, Pp alone may be misleading.

The higher the Pp value...
...the smaller the spread of the system’s output. Pp is a measure of spread only. A process with a narrow spread (a high Pp) may not meet customer needs if it is not centered within the specifications.

If the system is centered on its target value...
...Pp should be used in conjunction with Ppk to account for both spread and centering. Pp and Ppk will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Ppk

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.

If the system is centered on its target value...
...Ppk should be used in conjunction with the Pp index. If the system is centered on its target value, Ppk and Pp will be equal. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Pr

The Pr performance ratio is used to summarize the actual spread of the system compared to the spread of the specification limits (upper and lower). The lower the Pr value, the smaller the output spread. Pr does not consider process centering.

When the Pr value is multiplied by 100, the result shows the percent of the specifications that are being used by the variation in the process. Pr is calculated using the actual sigma (sigma of the individuals) and is the reciprocal of Pp. In other words, Pr = 1/Pp.

Cp

The Cp index is used to summarize a system's ability to meet two-sided specification limits (upper and lower). Like Cpk, it uses estimated sigma and, therefore, shows the system's potential to meet the specifications. However, it ignores the process average and focuses on the spread. If the system is not centered within the specifications, Cp alone may be misleading.

The higher the Cp value...
..the smaller the spread of the system’s output. Cp is a measure of spread only. A process with a narrow spread (a high Cp) may not meet customer needs if it is not centered within the specifications.

If the system is centered on its target value...
Cp should be used in conjunction with Cpk to account for both spread and centering. Cp and Cpk will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Cpk

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.

If the system is centered on its target value...
...Cpk should be used in conjunction with the Cp index. Cpk and Cp will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

Cpm

The Cpm index indicates how well the system can produce within specifications. Its calculation is similar to Cp, except that sigma is calculated using the target value instead of the mean. The larger the Cpm, the more likely the process will produce output that meets specifications and the target value.

Cr

The Cr capability ratio is used to summarize the estimated spread of the system compared to the spread of the specification limits (upper and lower). The lower the Cr value, the smaller the output spread. Cr does not consider process centering.

When the Cr value is multiplied by 100, the result shows the percent of the specifications that are being used by the variation in the process. Cr is calculated using an estimated sigma and is the reciprocal of Cp. In other words, Cr = 1/Cp.

[This message has been edited by Marc Smith (edited 15 February 2000).]

L

#### Laura M

I'm not Don, but I'll open the door for Don to agree or disagree with what I've understood the difference to be

I think the key is the fact that the estimated std. dev used for Cpk, being R-bar/d2 from the control chart, and the PPk be an actual std. dev. calculated on 100 individuals. R-bar/ d2 only takes into account "within subgroup variation", not "between subgroup variation" which the Ppk calculation would. If "all" process variation is present within the subgroups, then Cpk and Ppk will be very close. Selection of your sampling plan for the control chart are very relavant to the R-bar/d2 calculation.

Picture a process which has extremely tight within subgroup variation, but due to other circumstances, has variation - possibly acceptable variation...when the room warms up, tool wear, batches of raw material, etc. 5 consecutive pieces may have a Range near zero, but during a 20 day period x-bar's move. R-bar /d2 is not a good estimate of overall process variation. I never calculate a Cpk or Ppk without looking at the control chart. Moreso than looking at the control chart, you need to know how the numbers are generated to determine appropriate "rational subgroups."

#### Don Winton

##### Inactive Registered Visitor
Laura,

You are correct. I have been on a ten-week hiatus while I sorted some things out. I am now back at work and trying to get caught up.

Personally, I find all this Cpk/Ppk stuff too confusing. For almost 15 years, the 'C' values were working just fine, then all this 'P' stuff popped up. I ignore the 'P' stuff and just keep with what I know. This is my take on the difference.

In 1990, while a member of the Technical Staff at Hughes Aircraft Company, a gentleman named Suozzi wrote an excellent paper on process capability. It is the basis for the CPK.PDF paper of mine posted here at the PDF zone.

For Cpk calculations to be used, an unbiased estimate of sigma should be used. Calculations for Cpk use either the sigma symbol of the word "sigma." More accurately, the symbol sigma-hat should be used, since this is the graphical symbol for an unbiased estimate of sigma. Typically, sample sizes greater than 60 give an unbiased estimate of sigma so close to the actual sigma that correction factors are not needed. When the sample size is less than 60, a correction factor (designated C4) is used to obtain a more accurate estimate of sigma (this is explained in Souzzi's paper as well as mine and is based on a derivation from Grant and Leavenworth). Since the inception of Cpk, all the work I have seen has used either sigma or sigma-hat as the representation of the standard deviation.

Going back to basic stats for a moment, the calculation for standard deviation has either one of two possibilities in the denominator: n-1 or n. In school, when you knew the data for the population, you used n. When you did not know the data for the population, you used n-1. The two different types of standard deviation were designated by either sigma (population) or a lower case s (sample). Thus, just using simple logic, you would assume that Cpk is calculated for the population.

I asked the question here once what the AIAG used (I stay as far away from their stuff as I can. Gives me headaches), as the symbol in their method of calculating Ppk and the answer was the lower case s. Thus, it would seem to indicate that Ppk is the process capability for a sample and Cpk is the process capability for the population.

For process capability taken from a control chart, the same rules apply. If you have readings on more than 60 samples, a correction is not needed, less than 60, it is. Personally, I do not like this method and prefer to calculate process capability based on observed data, not recorded data. But, that is just my preference.

And I still think all this is too confusing.

Regards,

Don

[This message has been edited by Don Winton (edited 29 March 2000).]

#### David McGan

##### Inactive Registered Visitor
Now that certainly clears everything up! Thanks.

C

#### chuy sanchez

yea!!

is too confusing, only that i´m sure on ppap requirements gm wants to use a ppk > 1.67 or Cpk> 1.33.

so when i need to re-ppap my parts for some reason i presented to them ppk and guess what ???

they are happy !!!!

bye

#### Marc

##### Retired Old Goat
Staff member
By the way, Don, your paper CPK.pdf has been downloaded 525 times so far this month alone.

#### Don Winton

##### Inactive Registered Visitor
By the way, Don, your paper CPK.pdf has been downloaded 525 times so far this month alone.

I am glad some are finding it useful. Or at least, I hope they are.

Regards,
Don

[This message has been edited by Don Winton (edited 29 March 2000).]

#### Don Winton

##### Inactive Registered Visitor
Main Entry: ca·pa·bil·i·ty
Pronunciation: "kA-p&-'bi-l&-tE
Function: noun
Inflected Form(s): plural -ties
Date: 1587
1 : the quality or state of being capable; also : ABILITY
2 : a feature or faculty capable of development : POTENTIALITY
3 : the facility or potential for an indicated use or deployment {the capability of a metal to be fused} {nuclear capability}

As I have re-read this and other posts concerning process capability, it occurred to me that an assumption is being made that just ain't so:

Process capability is not a metric. But, it appears that is the assumption. The process capability index is a calculation made based upon data collected that determines if the process has the ability or potential[/b] to produce conforming product. There is no guarantee the process will do so.

Regards,

dWizard

#### Don Winton

##### Inactive Registered Visitor
Received my copy of Juran's Quality Handbook, 5th Edition a couple of days ago and found this of interest:

Ppk = Cpk-hat

The handbook defines Cpk as process capability and Ppk as process performance (Page 22.18).

Also, page 22.18 uses sigma in the denominator, but page 22.20 uses 's'.

Regards,

dWizard