Quality & Quantity 38: 95–111, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Normal Approximation to the Distribution of the
Estimated Yield Index S
W. L. PEARN
Department of Industrial Engineering & Management, National Chiao Tung University;
Department of Communication Engineering, National Penghu Institute of Technology;
Department of Applied Mathematics, National Chung Hsing University, Taiwan, R.O. China
Abstract. Process yield is the most common criterion used in the manufacturing industry for measur-
ing process performance. A measurement index, called S
, has been proposed to calculate the yield
for normal processes. The measurement index S
establishes the relationship between the manufac-
turing speciﬁcations and the actual process performance, which provides an exact measure on process
yield. Unfortunately, the sampling distribution of the estimated S
is mathematically intractable.
Therefore, process performance testing cannot be performed. In this paper; we consider a normal
approximation to the distribution of the estimated S
, and investigate its accuracy computationally.
We compare the critical values calculated from the approximate distribution with those obtained
using the standard simulation technique, for various commonly used quality requirements. Extensive
computational results are provided and analyzed. The investigation is useful to the practitioners for
making decisions in testing process performance based on the yield.
Key words: critical value, process yield
Process yield has longtime been a standard criterion used in the manufacturing
industry as a common measure on process performance. Process yield is currently
deﬁned as the percentage of processed product unit passing the inspection. That is,
the product characteristic must fall within the manufacturing tolerance. For product
units rejected (nonconformities), additional costs would be incurred to the factory
for scrapping or repairing the product. All passed product units are equally accep-
ted by the producer, which requires the factory no additional cost. For processes
with two-sided manufacturing speciﬁcations, the process yield can be calculated as
%Yield = F (USL) − F (LSL), where USL and LSL are the upper and the lower
speciﬁcation limits, respectively, and F (·) is the cumulative distribution function of
the process characteristic. If the process characteristic follows the normal distribu-
tion, then the process yield can be alternatively expressed as Yield% = (USL −
µ)/σ ] − [(LSL − µ)/σ ], where µ is the process mean, σ is the process standard
deviation, and (·) is the cumulative distribution function of the standard normal
distribution N (0, 1).