# Process Standard deviations and equivalence of Cp & Cpk

A

#### anrappa

By principle of natural tolerancing, the product tolerances are to be so selected that they lie within +/- 3SIGMA of the mfg process

My questions are :

1. Is a table / database giving the standard deviations of various processes(specifically machining processes)available.If yes, could somebody give me the reference/links.

If no, do we have to arrive at the standard deviation of various processes only thro analysis of past data.Could sombody clarify.

2. It is said that cpk = 1.67 implies cp = 2.00. How is this arrived at.

Regards:

D

#### Dave Strouse

Don't believe it exists

(1) I don't think you will be successfull in finding any tables for process standard deviations. There are so many factors, I can't imagine they would be tabulated.

How are you going to use it anyway? I believe you will need to develop this data yourself.

(2) WHO says.
I don't understand what in the world is meant.
Cp is (USL-LSL)/6s.
Cpk is closest or minimum of (USL- xbar/3s, Xbar-LSL/3s). The relationship of Cp to Cpk is Cpk =Cp(1-k) where k (|m-u|/(USL-LSL)/2 and m = (USL+LSL)/2 or the midpoint. But 0 to 1 is the range for k. So Cpk can be almost any value in relation to Cp.
The statement as I read it says that a Cpk of 1.67 always has Cp of 2. One counter example should disprove this.
Consider where LSL is 0 , USL is 10 sigma . Midpoint of spec is at 5 sigma. Cp is 10/6 or 1.67. Since a distribution could be centered on the mean i.e. k=0, we also have Cpk of 1.67.
Let's approach it another way. When will cp of 2 equal cpk 1.67?
From the formula above 1.67 = 2(1-k) or k =0.165.
So only at the k=0.165 value will the statement be true. Don't see how it implies anything remarkable.

A

#### Atul Khandekar

anrappa,

There are no 'standard' values for process std. deviations for manufacturing processes. In any case if you set the tolerance limits 'within' your +/- 3 Sigma limits what would be the process capability? Tolerance represents Customer's voice and process spread is the voice of the process. You will have to study your individual process variation and review tolerances only when it is just not possible for the process to produce within tolerance. I will quote from Grant:

"..specification limit is, of course, the service need of the part. This is not primarily a statistical matter....It often happens that the service needs can be judged more accurately with the aid of statistical methods....There is no use specifying desired tolerances on any quality characteristic without some prospect that these can be met...."

You could determine the variaton by conducting experiments and probably tell the designers what you can achieve. If you best effort does not meet the service requirement, I suppose you'll have to look for a better process.

Cpk=1.67 DOES NOT automatically imply Cp=2. This should be apparent from the formulas used to compute these indices as well as from a simple diagram that explains Cp/Cpk. Look at :
http://www.symphonytech.com/articles/processcapability.htm

Cp is (Spec width/process width).This does not take into account the centering of the precess. Cpk does that. (As Dave has already explained)

I think you are confusing this with something else -like a six-sigma process has Cp=2 etc?

-Atul.