## Abstract

Statistical Process Control monitoring of the ratio Z of two normal variables X and Y has received too little attention in quality control literature. Several 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 their proportions (ratios) within a product. Tables about the statistical performance of these charts are still not available. This paper investigates the statistical performance of a Phase II Shewhart control chart monitoring the ratio of two normal variables in the case of individual observations. The obtained results show that the performance of the proposed chart 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 T_{2}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 performance. An illustrative example from the food industry details the implementation of the ratio control chart.

Original language | English |
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Pages (from-to) | 1361-1377 |

Number of pages | 17 |

Journal | Quality and Reliability Engineering International |

Volume | 30 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Dec 2014 |

## Keywords

- Average run length
- Ratio
- Shewhart control chart
- Standardization
- Statistical Process Control