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NON normal data

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  • #27671

    Adam
    Participant

    according to wheels & chambers, One does not need to transform a non normal data  into normal using box cox in minitab, when making control charts. The reason being because “control charts work well in situations where data are not normally distributed. They give an excellent demonstration of the performance of control charts when data are collected from a variety of non-symmetric distributions.”
    could anyone throw further light on this. Is this true and if not why???
     

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    #68070

    denton
    Participant

    The best treatment I’ve seen on that issue is by Don Wheeler, “Normality and the Process Behavior Chart”.  He did an empirical study of 1143 different distributions, U shaped, J shaped, mound shaped, Webull, Burr and probably some others I don’t remember.  The long and short of it is that Control Charts work just fine with non-normal data.  Your limits won’t be 99.7%, but, in the very worst case, they will be about 98%, which will lead you to the correct conclusion much much more often that it will mislead you.
    The book has some other gems in it that I wouldn’t necessarily connect to Control Charts…. great reading.

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    #68074

    boettler
    Member

    I think the main point that Wheeler and Chambers were trying to make is that one should not shy away from using control charts just becasue the data are not from a normal distribution.  And I agree with this.  But I would also argue that control charts work best for normal data, and if you can transform your data to make it normal, you should do so. 
    With normal data, you know what your false alarm rates are, and they tend to be pretty reasonable.  With other distributions, the control charts may be overly sensitive to the natural random variation in the data or not sensitive enough to shifts in the process.  For example, I tried out some random data in MINITAB using a chi-squared distribution with 2 degrees of freedom.  Both the X-bar and R charts flagged too many out of control points, especially the R chart.  Also, all the out of control points were on the high side, indicating that the lower control limit was too low to detect shifts in that direction.  This is typical of a skewed distribution.  Applying the Box-Cox transformation eliminated the problem.  As another example, look at the uniform distribution in the Wheeler and Chambers book.  Clearly you would be better off with narrower control limits.
    Rob

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    #68083

    Gerry
    Participant

    Adam,I am not a statistician, although I’ve completed Black Belt certification.I believe the robustness of the control charts to non normal data and infact many statistical tests are based around the properties of the central limit theorem. Essentially by sampling data and performing analysis on the sample means the effect of non normality is minimised. Try generating a binomial distribution and sampling the data. The means are not significantly different and the sample sigma is a function of the population sigma divided by the square root of the sample size.

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    #68087

    Ken K.
    Participant

    Anytime you are using means (as in an X-bar chart) you can feel comfortable that the means will be nearly normally distributed, especially if the subgroup sample size is larger. The Central Limit Theory says that sums of random variates, regardless of their distributions, will tend toward a normal distribution as the number of samples comprizing the sum get larger. That is, means tend toward normality regardless of the underlying population distributions as the sample size increases.
    See http://davidmlane.com/hyperstat/A14043.html
    Control charts for individuals charts DO assume normaility, so you need to be very careful and make sure the incoming data ae normal. Another way to construct the control chart limits for non normal data would be to use nonparametric percentiles (if you had a very large sample to base them on).

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    #68093

    denton
    Participant

    I think most of what has been said is right, but I do have to dissent on some points.
    It’s true that Control Charts work better if you have normal data.  And it’s true that non-normal data will cause more false alarms.  It is also true that using an Xbar and R chart will tend to normalize the data, and is a good solution.
    Where I disagree is that you need normal data for an I&MR chart.  Just for fun, I ran an I&MR chart on 1000 data from a Chi Square distribution with 2 df.  I got 24 out of limit points, which is consistent with Wheeler’s finding of about a 98% limit, worst case.  Then I ran 1000 groups of 5.  Sure enough, the out of limit cases dropped to 13, just as it should.  I don’t think 24/1000 is an unacceptable number of false alarms, though I’ll agree that 13/1000 is better.
    I think that Wheeler’s point is that the error from non-normal data will tend to be on the conservative side.  You’ll have more false alarms, but you won’t miss signals.  In most cases, that’s an acceptable compromise. 

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    #68095

    John P.
    Participant

     
    Another point to consider is committing Type I and Type II errors.  If non-normal data is analyzed using an Individual Control Chart they you will get a lot of false positives i.e. Type I errors.  When the process is adjusted then your tampering and making the process go out of control.
    Another solution to for individual non-normal data is to use EWMA charting.  Using an lambda of 0.4 “provides a control chart with approximately the same statistical properties as a traditional X-bar chart combined with the run tests described in the AT&T Statistical Quality Control Handbook” (Pyzdek, The Six Sigma Handbook pg 449).  A lower lambda will reduce the sensitivity.

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    #68105

    Ken Myers
    Participant

    Denton,Range data are not normally distributed. Therefore, the number of false alarms would be higher than that found on the average chart.

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    #68297

    kbower50
    Participant

    The arguments for the robustness to Normality are associated with Tchebycheff’s inequality, as is discussed in Shewhart’s original discussions on control charting.  It’s true that the CLT will allow for approximate Normality, hence the false alarm rates frequently referred to would be valid in such an instance (importantly, related to independence also.)    I would be concerned with discussing such false alarm rates in the context of I-MR charts in the presence of non-Normality as the CLT, by definition, couldn’t assist since n=1.

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