Deming’s red bead experiment and mechanical sampling

The red bead experiment (RBE) is very well‐known in total quality management. It was designed by Dr W.E. Deming to show managers several lessons in management and sampling. The RBE has evolved over several decades and now has an accepted standard format. The RBE also serves as an introduction to control charts which are an important tool in statistical quality control. Surprisingly very little is known about the statistical properties of the RBE. This is most likely due to the fact that the RBE involves mechanical sampling and so is much more complicated than probability sampling. New methods are given for statistical analysis when mechanical sampling is involved (e.g. mechanical sampling of non‐uniform output in a manufacturing environment). This approach has specific applications to control charts which are a major tool in quality control. In a rigorous analysis, it is shown that p, the sample proportion of red beads in the RBE, is an unbiased estimator of the RBE process defect rate. Expressions are also found for the variance of p, enabling the calculation of control chart limits. Some of the techniques used can be applied to other mechanical sampling situations which may at first viewing seem intractable.