A new process capability indexCs, is proposed which extends the most useful index to date, the Pearn-Kotz-JohnsonCpmk, by not only taking into account that the process mean may not lie midway between the specification limits and incorporating a penalty when the mean deviates from its target, but also incorporating a penalty for skewness. The new index is specifically designed to help control skewness-prone manufacturing processes typified by a hole-drilling process. (The distribution of hole diameters produced by an in-control drilling process is usually near-normal but truncated on the left, at the drill diameter, and slightly skewed to the right. The characteristic behavior of a worsening hole-drilling process is a simultaneous increase in mean, variance and skewness, with little change in the lower percentiles of the distribution.) An estimatoris proposed and its bias and variance is studued bt simulation. The simulation studies show taht the new estimator compares favourably witheven when skewness is not present. Hence the use ofis advocated for monitoring near-normal processes where loss of capability typically leads to asymmetry.