# One Sided Tolerance Distribution(s)

#### Uchiha

##### Inactive Registered Visitor
Hello everyone,

First many thanks for the incredible job done here. This is my first post, but this forum already helped many times before today...

I know the question of process capability for one sided tolerance characteristics was discussed before (maybe more than once). However, there is still something bothering me that I can't figure out. It's not only about how to calculate the capability though...

The following are my personal conclusions, so please correct me if I am wrong:
There are two types of characteristics with one sided tolerance:

1- With natural "limit": for example perpendicularity or flatness. These cannot have a value less than 0 and therefore have only the upper tolerance. These characteristics should follow a folded normal distribution. Right?

2- Without natural "limit", but rather with only one actual specification: for example tensile strength which should not be less than X MPa (lower tolerance), but that can go up to whatever value possible. These characteristics should follow a normal distribution, right? Or is there any other specific distribution for these cases (one sided normal distribution for example)?

Is the Cpk calculated the same way in both cases anyway? And is the calculation based on the same dispersion used for normal distribution (3*sigma)?

Once again, thanks for the great job.
Uchiha

#### Stijloor

##### Super Moderator
Staff member
Super Moderator
A Quick Bump!

Can someone help?

Thank you very much!

#### Miner

##### Forum Moderator
Staff member
Super Moderator
There are two types of characteristics with one sided tolerance:

1- With natural "limit": for example perpendicularity or flatness. These cannot have a value less than 0 and therefore have only the upper tolerance. These characteristics should follow a folded normal distribution. Right?
Not necessarily. If the "natural limit" is artificial, such as Position where you are using the (polar) absolute value of the x/y coordinates that may actually be negative, or flatness that does not distinguish between concave/convex then you will likely have a folded distribution. However, if the natural limit is real as you would get by machining to a hard stop, you would be more likely to get a lognormal or similar distribution.

2- Without natural "limit", but rather with only one actual specification: for example tensile strength which should not be less than X MPa (lower tolerance), but that can go up to whatever value possible. These characteristics should follow a normal distribution, right? Or is there any other specific distribution for these cases (one sided normal distribution for example)?
The tensile strength distributions that I have worked with have all tended to be skewed. By theory, they should follow a smallest extreme value (i.e., weakest link) distribution.

Is the Cpk calculated the same way in both cases anyway? And is the calculation based on the same dispersion used for normal distribution (3*sigma)?
No. You should perform a non-normal capability study.

Some would advise that you transform your data using a Box-Cox transform, but I do not recommend this. I work with the native distribution.

#### bobdoering

Trusted
1- With natural "limit": for example perpendicularity or flatness. These cannot have a value less than 0 and therefore have only the upper tolerance. These characteristics should follow a folded normal distribution. Right?
Typically, through distribution fitting, I find the most typical disttributions for one-sided physical limit tolerances to be beta or Weibull distributions.

2- Without natural "limit", but rather with only one actual specification: for example tensile strength which should not be less than X MPa (lower tolerance), but that can go up to whatever value possible. These characteristics should follow a normal distribution, right? Or is there any other specific distribution for these cases (one sided normal distribution for example)?
There really is no location-specific capability (as in Cpk or Ppk) for one-sided distribution, as the whole point of their existence is to determine if the mean of the distribution is near the center of the tolerance. However, the center is not the goal of a one-sided tolerance, so the point is moot and nonsensical. You can use Cp or Pp.

Is the Cpk calculated the same way in both cases anyway? And is the calculation based on the same dispersion used for normal distribution (3*sigma)?
No, Cpk is not applicable to either. Cp or Pp can be calculated via transformation.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Why even do distribution based Cp/Pp/Cpk/Ppk? why not just plot your data in time sequence against the specificaiton limits? this will tell you how 'capable' you are. If you are producing defects, a simpel defect rate calculation should be sufficient to quantify the 'incapability'. What infromation will you derive from a "CPpk" calculation? what actions will it drive? Is it really worth all of the time spent figuring out how to calculate it? isn't it better to spend your time in actually improving the capabilty? just a thought...