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# Statistical Process Control via the Subgroup Bootstrap

Summary: The most commonly used techniques in statistical process control are parametric, and so they require assumptions regarding the statistical properties of the underlying process. For example, Shewhart control charts assume that the observations are independent, and that the statistic of interest is normally distributed. These assumptions are often violated in practice; for example, the distribution of the variable being measured may be strongly skewed or may fail a test for normality. In such cases the control limits, especially for small subgroup samples, may not be accurate. The bootstrap is a computer intensive resampling procedure that does not require a priori distribution assumptions. It was developed to find the distribution of a statistic when the distribution is not known. We first extend the bootstrap percentile method to include a series of subgroups, which are typically used in assessing process control limits. We show, via examples, how the subgroup bootstrap is used to assess process control limits for X and S2 charts. Via simulation, we then empirically compare the subgroup bootstrap and parametric methods for determining process control limits for a quality related characteristic of a manufacturing process under various conditions. The results show that bootstrap methods for X and S2 control charts generally achieve comparatively better control limit estimates than standard parametric methods, particularly when the assumption of a normal process distribution is not valid. The subgroup bootstrap is easily implemented on a personal computer as a general methodology for statistical process control, and hence, is a potentially useful pragmatic quality improvement tool.

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**Topics:**Statistical Process Control (SPC)**Keywords:**Statistical process control (SPC),Shewhart control chart,Bootstrap methods,Non-normality,Control limits**Author:**Seppala, Tomi; Moskowitz, Herbert; Plante, Robert; Tang, Jen**Journal:**Journal of Quality Technology