Estimating Percent within Limits for Skewed Populations

Computer simulation, using samples of size 3, 5, and 10, was used to evaluate the percent within limits (PWL) estimates for populations with skewness coefficients from 0.0 to ±3.0. The average bias and variability were then determined for the 10,000 PWL estimates. The analyses show that even a moderate amount of skewness in the underlying population can affect both the accuracy (bias) and the variability of individual lot PWL values. The amount of bias increased as the amount of skewness increased, and the bias also increased as the sample size increased. The results show that with two-sided specifications the bias varies not only in its magnitude but also in its sign, that is, positive or negative, depending on the split of defective material outside the lower and upper limits (the PD_L/PD_U split). The amount of variability also increases as a greater percentage of the defective material is on the long tail of the distribution. Increased variability means that errors in individual lot estimates will be larger and thereby create a greater spread in the pay factors on a project. If skewness is present, the population cannot be identified only by its PWL value. It will also be necessary to consider what percentage of the defective material is outside each of the specification limits. The analyses indicate that erroneous results for a population's PWL value can be obtained if the population is only moderately skewed. For this reason it is important to monitor process data to ensure that they are from an approximately normal population.