Estimated Uncertainties in Control Limits

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    Stan Alekman

    Like all statistics, they are uncertain and are often stated as an interval.
    Since the group of samples tested and average value charted is assumed to have within-group variance of zero, and for other reasons as well, the formulae that are used to provide point estimates of control limits have uncertainty associated with them. How does one estimate the uncertainty in control limits? 


    Dave Strouse

    Stan –
    I’m a little confused by your post.
    I’ve not seen control limits for control charting ever expressed as an interval. Could you provide an example?
    There is no assumption of zero variance in the subgroups for calculating control limits. Again, can you name a source for this?
    Control limits are calculated by estimating the within subgroup variance by using the within subgroup range or the within subgroup standard deviation and modyfying this estimate by the control chart factors to give an economic balance between alpha and beta risk.
    I’m not sure why you would even want to know the confidence interval around a control limit. It’s application is entirely empirical. I suppose you could calculate worst case risks, but 75 years of experience says that estimating the limit as above and reacting when it is exceeded is good practice. What would you do differently if you had a range for the limit?


    Neil Barrass

    A related question came up in my company. We review our CL’s periodically and recently I was asked about whether it was worth changing the CL’s in the SPC system when the change was of the order of 1%, ie a very small change.
    My response was that CL’s are empirical limits at which, if exceeded, indicate that statistically something MIGHT have changed.  Thus small changes to CL’s isn’t important but I couldn’t answer the “how big is important?’ question.
    Any thoughts?


    Erik L

    A modification to the control limits should only be considered when there is a shift in the process.  Typically, a violation of the Western Electric rules for a run of points in a row above or below the center line is indicative of a process shift and would signal a need to recalculate the control limits.  A point outside of the limits would not indicate a need to recalculate the limits.  Why would we bother to create them then?  The control charts should be relied on to guide us in modifying the limits or keeping them the same.
    A point out of control should be regarded as a significant shift in the voice of the process.  The within subgroup variation, yielding Rbar, and the appropriate subgroup constant are designed to put the control limits +/-3 standard deviations.  The apha risk of interpreting noise, as if it was a signal, is still there, but you should consider all points outside as a potential chance to improve the process.  Eventually, with a stable process and enough data you will fall into the alpha region of the distribution and find points outside of the control limits.


    Neil Barrass

    I wasn’t referring to a reaction to OOC. It’s just that as the (stable)process runs and we gather more data for any period, we will calculate  slightly different CL’s and Centre Lines (All processes drift around a bit, both mean and variability, that’s where 1.5s came from). These variations are quite small, so the question was “When to reset limits?” The issue is whether there has been a statistically significant change and whether this change has practical significance.

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