CI on capability question

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    Often I found that Cpk (Ppk, Cp Pp…) tends after calculated sometimes seams to be regarded as the absolute truth.
    In order to point out that it is actually Cpk-hat I often use CI function in MiniTabs Capability graph to show the interval. (and some other excel based tricks).

    However, not everyone has the benefit of having at statistical analysis program at hand, but almost everyone have Excel. So what I have try to do are calculate the same thing in excel.

    Basically what I have done are to calculate CI for X-bar and SD, so far so good.

    Now to the question. Not holding a PhD in statistics (just an ordinary BB, probably slightly under average, Stanine 4 or something ;-)) this made me thoughtful how to do this correct.

    My initial idea was to take the maximum/minimum and combine them in the outermost positions to get a “worst case” for the capability calculation (I haven’t found any documentation how the calculation are made). Even before I pressed enter it did not felt ok…Buggers…

    Somewhat like a skibum friend of mine. Desperately needed to ski in the middle of summer. Analysis: pair of skis, nice slope to the village (Road downhill, nicely serpentined) some old skateboard tracks and wheels. Solution: bolt skateboard wheels to ski, interact bum/ski/slope. Result: Nice skiing in the middle of summer. Also so far so good…until he did realize a slight difference, however important, between this and normal skiing. Wheels doesn’t break the same way as ski on snow… Moral: a beautiful and simple idea does not always yield all the way

    So before I find my self doing 30-something with only painfully ways to break on capability CI issues, anyone of more knowledge than me who can shine some light on this?

    (For Cpk SD in excel I am using s-bar)



    there is a relatively simple formula that applies to the case of normally distributed population. It only calculates the lower limit on Cpk, but that should be all right as no one is really interested in the upper limit anyway.

    The formula is:

    lower limit = Cpk – Zalpha*SQRT((1/9n +Cpk*Cpk/2*(n-1))

    Zalpha is the z value associated to your confidence level – I guess it should be the one-sided value BTW.
    n is the sample size
    Cpk is the point estimate of your Cpk value – the one you calculated based on your sample.

    I hope this helps.

    P.S. the source is the book”Statistical Methods for Quality Improvement” by Ryan. It also has a different, more accurate formula but is is way too complex IMHO for practical use.




    Hi and thanks!

    You are probably right, that formula is accurate enough to prove my point about Cpk-hat.
    And I also agree on lower limit being the point of most interest.
    For the upper limit, the most practical reason for that might be:
    “Ok, so your process might be as bad as the lower limit, on the other hand I can be as good ass the upper. So why don’t you take some more samples an find out?”

    (I was tempted to write: I know what you are thinking, punk. “my Cpk are 1,33, but did I take enough samples?”. Now to tell you the truth, I didn’t look at your data in all the excitement. But this is the capability calculation, the most powerful calculations in ten world* and it can calculate your process right off, you’ve gotta ask yourself a question: “Do I feel lucky?” Well, do ya, punk?
    *) plenty may disagree on that it is ;-))

    But for arguments sake if I would like to calculate CI in the “correct” way what does interact with what? Difference between Cpk and Ppk? What I did feel bad about before pressing enter in excel was the mixture of probabilities occurring in the formula.

    Probability for x-bar AND s being in worst case at the same time

    For Cpk I use s-bar to calculate SDst, CI for s-bar?

    That’s were I started to feel a bit dizzy in my head.
    Yhea, I will probably use the formula described, just want be prepared fore the question:

    Hey!! There is a difference for the CI I the equation you taught us and from what MiniTab says! Why?

    Regards and thanks again




    “Hey!! There is a difference for the CI I the equation you taught us and from what MiniTab says! Why? “

    I would say that these are not “exact” formulas but approximations. There is a much more precise formula then the one I gave, but it is quite complex – it involves solving a non-linear equation for the lower limit of the CI. And of course other approximations might also be possible – Minitab is probably using a more involved one then the simple formula in Excel.


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