# Process Capability of variance

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• #31463

I have a process that gives me position data of a machined feature in terms of deviation from the nominal value. How do I calculate process capability of such a measure?

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#82989

Hi Rahul,
Your characteristic of interest is the deviation from target.  Each machined part will have one such measurement.  The first step in all capability studies is to check for process stability.  This means these deviations must be plotted on a control chart.  I would use an IX & MR chart if the production rate is low or if there is a long cycle time.  If this is a high-volume operation, then I would recommend an X-bar, R chart.
Once you have achieved stability, the average deviation from target (mu) for this process is estimated from the centerline of the IX (or X-bar) chart.  Hopefully, mu-hat is close to 0!
mu-hat = X-bar  or   mu-hat = X-double-bar
The short-term process standard deviation (sigmaST) of these deviations from target is estimated by dividing MR-bar (or R-bar) by the appropriate d2 factor for the subgroup size used for the chart.
sigmaST-hat = MR-bar / d2   or   sigmaST-hat = R-bar / d2
If you prefer, the long-term process standard deviation (sigmaLT) can be estimated by first calculating the overall sample standard deviation, S, from all the collected measurements.  Then divide S by c4, which is based on the total number of measurements in this study.
sigmaLT-hat = S / c4
Before estimating any of the “standard” capability indexes, you should check to make sure the measurements have close to a normal distribution.  This check can be done with either a goodness-of-fit test or by plotting the measurements on normal probability paper.  If the normality assumption is met, you can procede with the next paragraph.  If the normality assumption is not met, then you must use a measure of capability that is designed for non-normally distributed data (this would be a good topic for another complete discussion!).
To estimate process capability for normally distributed data, you can use Cp, Cpk, Pp, or Ppk.  In your situation, you may not have a lower spec limit (LSL), which means you cannot estimate either Cp or Pp.  Note that “0” is not the LSL, it is just the lower bound for the deviation-from-target measurement.
Cp-hat = (USL – LSL) / (6sigmaST-hat)
Pp-hat = (USL – LSL) / (6sigmaLT-hat)
Cpk-hat = Minimum[(mu-hat – LSL) / (3sigmaST-hat) , (USL – mu-hat) / (3sigmaST-hat)
Ppk-hat = Minimum[(mu-hat – LSL) / (3sigmaLT-hat), (USL – mu-hat) / (3sigmaLT-hat)
If there is no LSL, then these become:
Cpk-hat = (USL – mu-hat) / (3sigmaST-hat)
Ppk-hat = (USL – mu-hat) / (3sigmaLT-hat)
I hope your process has high capability!

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#83038

Ross, Thanks for the information.
I cannot divide my data into rational subgroups for an Xbar or any other quality chart. Would using the ‘Cpm’ measure in minitab be a good indicator for process capability. Since my target is zero, wouldn’t Cpm suffice for this purpose. Would greatly appreciate your coments on this. thanks.

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#83043 Robert Butler
Participant

Hole location along with a host of other characteristics such as taper, flatness, squareness, shrinkage, etc has the property that it is bounded by some physical limit.  In the case of hole location, you can’t have a location that is less than zero.  As was noted in another post, your limit is zero-perfect location every time.  As you move toward the limit the distribution of the measurements gets more and more skewed.  The lower tail cannot go below zero but the upper tail has no upper limit.  For distributions of this type Cpk does not apply since the distribution violates the fundamental  tenet of normality.
In order to compute an equivalent Cp you will have to take your data, plot it on normal probability paper, identify the values corresponding to .135 and 99.865 percentiles (-3 and +3)   The difference of these two values will be your equivalent 6 sigma spread.  Your equivalent Cp will be the tolerance (USL-LSL) divided by the equivalent 6 sigma spread.  You may very will find that this number does a poor job of characterizing the performance of the process as you know it.  In that case it might be better to express your process in terms of percent nonconforming.
You should check Chapter 8 -Measuring Capability for Non-Normal Variable Date in Bothe’s book Measuring Process Capability, for additional information and caveats.

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#83046

Hi Rahul,
Unfortunately, Cpm has the same assumptions as the other capability measures, namely, stability and normality.  In addition, the calculation of Cpm requires both an upper and a lower print specification.  As mentined previously, there is no LSL in this situation.
The best that you can do with this data is to first estimate the long-term standard deviation as shown in my previous post.  Then, check for normality.  If the normality assumption is met, estimate the Ppk index.  This will give you an idea of process capability, assuming that the process output was stable when these parts were made.  This is a big assumption and must be stated right next to the Ppk index in all your written reports and prominently mentioned in all oral reports.
Next time, get your supplier to at least measure the parts in production order.  With these data, you can construct a control chart and properly estimate process capability.
Hope this helps.

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#83148

Doesn’t plotting my data on normal probability paper amount to converting my data to normal data, by using a box-cox transformation? While Cp would be a good measure to use during product development ,it would be wise to look at it (or Pp values) over a manufacturing run? I’d rather look at the Cpk or the Ppk values if I was evaluating long term, knowing my current process was stable.
As to non-conforming products, there are currently no such items produced. However, to change the gauge plan to measure less frequently, I still need to know that any variation in the data over a long period of time will not lead to non-conformances.

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#83158 Robert Butler
Participant

Plotting data on normal probability paper does not transform the data in any way.  All that you are doing when you plot the cumulated distribution on the paper is giving yourself a sense of the location of your case percentages.  The straight line on normal arithmetic probability paper is primarily for reference purposes-that is, if your plot falls on the straight line there is a pretty good chance that your data is normal.
If you want to check this run a normal plot of the following data:
1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,14
Then take the log of this data and run a normal plot of the logged values.  The first plot will form an arc relative to the straight line whereas the second will hold fairly close to the straight line.
As for equivalent Cp if you are interested in long term then there is an equivalent Phatp which is expressed as
Tolerance/(Equivalent 6 sigma Long Term Spread)
where the equivalent 6 sigma is computed by taking the difference of xhat(99.865) amd xhat(.135). pp. 434-441 of the Bothe book provides additional information.

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