Statistical analysis of deadband-deviation rules in quality control

The paper is concerned with the statistical analysis of rules designed to handle deviations from a prescribed tolerance interval, named a deadband, around a desired specification for a production process. The rules are ostensibly designated to recalibrate the process at the intended specification and as such reflect a certain interpretation of the deviation from it-either a machine shift or a random error or any combination thereof. The true nature of the deviations, however, is always unknown. Statistically it is assumed that any actual measurement is symmetrically distributed around the last calibrated value of the production process and that the repeated calibratins only shift around the location of the said distribution but do not alter its shape. The analysis is done under the hypothesis that all deviations are indeed random errors and are not due to any machine shift, namely, a null hypothesis type background. The principal parameters engendered by the calibration rules are investigated. First it is proven that the repeated, and possibly frequent, adjustments to the production process do not fundamentally affect it in the sense that the expectation of any calibrated value remains the original specification of the process. This property is crucial since otherwise the calibration rules would have generated bias rendering them useless. Next an expression is developed for the variance of the calibrated values. Because of the prohibitive form of the latter it is necessary to resort to simulation in order to explore its behavior. The paper ends with a suggested plan for the study of the same parameters under alternative type hypotheses.