Degrees of freedom for XmR charts

[mattiase]

Hi,

I'm developing an automatic tool where we calculate the Cp, Cpk etc. for our production data. I would like also to present the confidence level for the Cp and Cpk calculations. I am using these formulas below from Minitab Technical Support Document: Confidence Intervals Cp Cpk (Minitab Knowledge Base ID 853: Technical Support Document Confidence Intervals Cp Cpk). It is always nice to compare with Minitab to see that everything is correct.

However, in these formulas the degrees of freedom(v) is stated as:
v=k-Rspan+1
example for a 100 samples, with 2 sample moving range this will be:
v=100-2+1 = 99 dof
actually exact same degrees of freedom as pk and ppk would use.

But, I've just studied the excellent book: "Advanced Topics in Statistical Process Control, 2nd Edition" by Donald J Wheeler, and in this book he clearly states that for the average moving range the degrees of freedom is approx (page 62 and Table 23):
v= 0.62*(k-1)
for the same example as above, this would result in:
v=0.62*99 = 61 dof, which is signicant lower than the formula Minitab uses and would affect confidence interval calculations a lot, actually make it wider.

The degrees of freedom is also discussed by Wheeler in the Quality Digest article in the June 97 issue by Donald J Wheeler: "How Much Data Do I Need?"

I am a bit confused here. Which is actually correct definition for degrees of freedom using average moving range?

Thanks,
Mattias

PS! This is my first post ever, so I wasn't able to insert links, but just google for my sources if you would like to study them. Use: "degrees of freedom moving range" and "Minitab 853". Please be kind...

 

[Steve Prevette]

Welcome to the Cove. Well, you made me aware of something I'd never seen before. I'm not sure this would really affect the results much. In X moving Range charts I've done, I've just used 2.66 times the average moving range for the X control limits. I'm rather surprised knowing Dr. Wheeler's other work that he would get into such apparent minutae, but I'm not a phD.

I don't use Minitab, but if you'd like to see the impact I'd suggest take the same set of date and run it each way and see, practically, if it really makes a difference.

 

[mattiase]

Hi again,

I did some test of a set of 100 production value on an electrical audio amplitude. 95% confidence interval inside the brackets.

With DoF=61 (according to Wheeler):
Cp= 2.06 [1.70;2.43]
Cpk=2.02 [1.66;2.39]

With DoF = 99 (according to Minitab):
Cp= 2.06 [1.78;2.35]
Cpk=2.02 [1.73;2.31]

Yes, there is a difference in the confidence interval. A bit to much to just ignore I think.

Any ideas of the definition of degrees of freedom? Degrees of freedom is important when interpreting the cp and cpk, that is obvious.

 

[Steve Prevette]

Actually, I would not consider those differences to be worth worrying about (IMHO).

 

[mattiase]

Yes, I agree, but it would still be very interesting that the correct definition of degrees of freedom is, though.

 

[Steve Prevette]

Yes, I agree, but it would still be very interesting that the correct definition of degrees of freedom is, though.

True. Generally speaking, degress of freedom represent the number of data values "used up" in estimating parameters for the analysis. For example, linear regression "uses up" two degrees of freedom in estimating the slope and intercept. This does lead to a person questioning what is the interpretation of a fractional degree of freedom . . .