Capability indices and the proportion of nonconforming items

For nonnormal data, there exists no standard calculation for the process capability. Many different indices were developed and described in the literature, but none of them predict the proportion of nonconforming items well enough. This paper starts with a sharp upper bound for the proportion of nonconforming items if the distribution is in the class of unimodal distributions. This class contains the Student's t, the logistic, the gamma, the ¿2, the exponential, the log-normal, and the Weibull distributions. For a given Cpk-value, the maximum proportions nonconforming for these parametric distributions are also determined and compared with the upper bound. These maximum proportions nonconforming are a first step to relate the process capability to the proportion of nonconforming items. Furthermore, two different capability indices for skewed data, developed by Munechika and Bai and Choi, are investigated on their ability to adapt to nonnormality. It turns out that for our selection of distributions, the Munechika capability index corrects the standard capability index Cpk too much and the Bai and Choi capability does not compensate the Cpk enough. Since this occurs for each of our skewed distributions, the true capability (based on the inverse normal distribution of the proportion nonconforming) lies in between these two capability indices. Therefore, we propose to calculate the maximum proportions nonconforming, the standard capability indices, and the indices of Munechika and Bai and Choi as a standard approach to determine the process capability. This approach maintains the simplicity of capability indices as a way to express a process in only a few numbers, and it indicates all aspects of the process capability. This method is illustrated in a real-life example