Design of a Phase II Control Chart for Monitoring the Ratio of two Normal Variables

In many industrial scenarios, on‐line monitoring of quality characteristics computed as the ratio of two normal random variables can be required. Potential industrial applications can include monitoring of processes where the correct proportion of a property between two ingredients or elements within a product should be maintained under statistical control; implementation of quality control procedures where the performance of a product is measured as a ratio before and after some specific operation, for example a chemical reaction following the introduction of an additive in a product and monitoring of a chemical or physical property of a product, which is itself defined and computed as a ratio. This paper considers Phase II Shewhart control charts with each subgroup consisting of n > 1 sample units. From one subgroup to another, the size of each sample unit, upon which a single measurement is made, can be changed. An approximation based on the normal distribution is used to efficiently handle the ratio distribution. Several tables are generated and commented to show the statistical performance of the investigated chart for known and random shift sizes affecting the in‐control ratio. An illustrative example from the food industry is given for illustration.