# stat question

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• #42203

Participant

hello
Cpk = min(Xbar-LSL/3sig,USL-Xbar/3sig)
my question is, why does the Cpk change is you assume diffrent distribution for the same data set, is it the standard deviation that is calculated diffrently or the mean or both from distribution to distrinution, i always thought that Xbar is always sum all/n, and Sd was always calculated the same way as the normal distribution sum(Xbar-Xi)^2/(n-1) please exaplain

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#133128

Robert Butler
Participant

Since the calculation of Cpk is predicated on the assumption that the data is normally distributed any changes to this assumption will result in changes in the Cpk values.
For computation of equivalent Cpk’s for non-normal distributions you should read Measuring Process Capability by Bothe – specifically Chapter 8 – Measuring Capability for Non-Normal Variable Data.
In these cases the equivalent six sigma spread is determined by plotting the data on normal probability paper and identifying the values corresponding to the .135 and 99.865 percentiles.  If you take the difference between the measured values at these two percentiles you will have your equivalent six sigma spread which you then use for computation of the equivalent Cpk.

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#133129

Participant

my main question is in the Cpk formulae, which perameter has diffrent equasion from distribution to distribution. thanks again

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#133132

Jered Horn
Participant

I probably should be more helpful and post the formulas for you, but I’m not…look them up.
Cpk usually is associated with short term process capability…a measure of process technology.  There are at least five ways to estimate short term standard deviation (Rbar, Sbar, Pooled, Average Moving Range, Median Moving Range).  All different formulas and they estimate the standard deviation within subgroups.
The formula you posted for standard deviation for “normal distribution” is actually the formula for variance.  If you take the square root of that, you get what is generally called the estimated long term standard deviation.  It takes into account variation between subgroups and is a measure of process control.  And to add more to the confusion, Minitab divides this standard deviation by an “unbiasing constant” C4(d+1), which is beyond my comprehension at this point in the morning.  You do, at least, have the option not to use it.
Hope that helps.

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#133133

Robert Butler
Participant

The standard deviation.
From Bothe pp.433 “For normal distributions 6sigma long term predicts the span required to produce the middle 99.73 percent of the process output and is one measure of potential capability (this sigma is computed as per the formula in your first post) But for the 6sigma long term spread to accurately represent the middle 99.73 percent of the process output requires the output to have a normal distribution. Thus, before making any decisions based on this measure of capability (in fact before even estimating it), the normality assumption must be verified.”
From this point in the book Bothe goes on to describe the method for determining the 6sigma long term spread which I outlined in my first post.  Thus for non-normal data you do not use the formula for computation of standard deviation – you determine it graphically and it will change from distribution to distribution.

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#133137

Peppe
Participant

If I well understand you are saying that one can calculate the same thing with different formulas, getting different results ? Is it correct ?
Rgs, Peppe

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#133139

Jered Horn
Participant

Peppe,
Are you trying to trap me into uncovering the fact that I am not a statisticiain?  I’ll admit that…proudly.
But, yeah, that’s what I’m saying.  The different methods of estimating short term standard deviation will give you slightly different results.  There are some “rules” that I didn’t mention.  The Rbar, Sbar, and Pooled methods require subgroup size >1.  The Average Moving Range and Median Moving Range are for subgroup size = 1.
I think the operative word here is “estimate”.  These are all ways to estimate the standard deviation of an entire population.
And BTW…I’m not sure my post answered the original poster’s question.  I think Mr. Butler took care (or IS taking care) of that.

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