Quality & Quantity 37: 443–453, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Distributional and Inferential Properties of the
Estimated Precision C
Based on Multiple Samples
W. L. PEARN
Department of Industrial Engineering & Management, National Chiao Tung University;
Department of Industrial Engineering, Da-Yeh University, Taiwan ROC
Abstract. Process precision index C
has been widely used in the manufacturing industry to provide
numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C
for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed
an efﬁcient test for C
based on one single sample, and showed that the test is the UMP test. In this
paper, we consider an unbiased estimator of C
for multiple samples. We show that the unbiased
estimator is the UMVUE of C
, which is asymptotically efﬁcient. We also consider an efﬁcient test
, and show that the test is the UMP test for multiple samples. The practitioners can use the
proposed test on their in-plant applications to obtain reliable decisions.
Key words: process precision index, unbiased estimator, UMVUE, asymptotically efﬁcient, UMP
test, p-value, power.
Process capability indices, which establish the relationships between the actual
process performance and the manufacturing speciﬁcations, have been the focus of
recent research in quality assurance and process capability analysis. Those cap-
ability indices, quantifying process potential and performance, are important for
any successful quality improvement activities and quality program implementation.
The ﬁrst process capability index appeared in the literature is the precision index
, which is deﬁned by Kane (1986) as:
where USL is the upper speciﬁcation limit, LSL is the lower speciﬁcation limit,
and σ is the process standard deviation. The numerator of C
gives the range over
which the process measurements are allowable. The denominator gives the range
over which the process is actually varying. The index C
was designed to measure