Statistical performance of a control chart monitoring the ratio of two normal variables

On-line Statistical Process Control (SPC) monitoring the ratio Z of two normal variables X and Y has surprisingly received too little attention in the quality control literature. Several potential applications dealing with monitoring the ratio Z can be found in the industrial sector, when quality control of products consisting of several raw materials calls for monitoring the stability of the proportions (ratios) of different components within a product. Tables about the sensitivity of these charts in the detection of one or more assignable causes are still not available. This paper investigates the statistical performance of Phase II Shewhart control charts monitoring the ratio of two normal variables in the case of individual observations. The obtained results show that the performance of the proposed charts is a function of the distribution parameters of the two normal variables. In particular, the Shewhart chart monitoring the ratio Z outperforms the (p=2) multivariate  control chart when a process shift affects the in-control mean of X or, alternatively, of Y and the correlation among X and Y is high and when the in-control means of X and Y shift contemporarily to opposite directions. The sensitivity of the proposed chart to a shift of the in-control dispersion has been investigated, too. We also show that the standardization of the two variables before computing their ratio is not a good practice due to a significant loss in the chart’s statistical sensitivity. An illustrative example from the food industry is detailed to show the implementation of these control charts.