Two new types of tolerance intervals for sample variances

The variance of a quality characteristic reflects the variability of a product or a process and its increase (or decrease) usually indicates a deterioration (or an improvement) in the quality of the product or the process. However, the traditional equal-tailed β-content tolerance interval (TI) for the sample variance has some limitations, such as the average coverage far from the target value β, the large standard deviation of the coverage, the insufficient accuracy when the number of subgroups or the subgroup size for constructing the TI is particularly small. In quality control applications, the reference dataset is often limited due to time or cost constraints, so this paper focuses on improving the accuracy level and proposes two new methods (one is to minimize the deviation between the average coverage and the target value β, and the other one is to centralize the coverage) to design TIs, and extends them from the frequentist perspective to the Bayesian perspective. The performance of the TIs is evaluated and compared using the average coverage, the standard deviation of the coverage, the accuracy level, and the difference between the lower and upper tolerance factors. The results show that both proposed TIs have an overall better performance than the corresponding traditional equal-tailed TI in both frequentist and Bayesian perspectives. Finally, a real example is used to demonstrate the design, the superiority, and the practical application of the proposed β-content TIs.