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The authors declare that they have no competing interests.
All authors contributed to the design of the project. AP set up and implemented the resampling algorithms and wrote the first draft of the manuscript. TW determined the mathematical proofs in determination of the maxima for average moving range. All authors contributed to later drafts and gave final approval to the manuscript.
The XmR chart is a powerful analytical tool in statistical process control (SPC) for detecting special causes of variation in a measure of quality. In this analysis a statistic called the average moving range is used as a measure of dispersion of the data. This approach is correct for data with natural underlying order, such as time series data. There is however conflict in the literature over the appropriateness of the XmR chart to analyse data without an inherent ordering.
We derive the maxima and minima for the average moving range in data without inherent ordering, and show how to calculate this for any data set. We permute a real world data set and calculate control limits based on these extrema.
In the real world data set, permuting the order of the data affected an absolute difference of 109 percent in the width of the control limits.
We prove quantitatively that XmR chart analysis is problematic for data without an inherent ordering, and using real-world data, demonstrate the problem this causes for calculating control limits. The resulting ambiguity in the analysis renders it unacceptable as an approach to making decisions based on data without inherent order.
The XmR chart should only be used for data endowed with an inherent ordering, such as a time series. To detect special causes of variation in data without an inherent ordering we suggest that one of the many well-established approaches to outlier analysis should be adopted. Furthermore we recommend that in all SPC analyses authors should consistently report the type of control chart used, including the measure of variation used in calculating control limits.