PROC CAPABILITY and General Statements 
Specialized Capability Indices
This section describes a number of specialized capability
indices which you can request with the
SPECIALINDICES option in the PROC CAPABILITY
statement.
The Index k
The process capability index k (also denoted by K)
is computed as
where m = (1/2)(USL + LSL) is the
midpoint of the specification limits, is the
sample mean, USL is the upper specification limit,
and LSL is the lower specification limit.
The formula for k used here is given by Kane (1986).
Note that k is sometimes computed without taking the
absolute value of in the numerator.
See Wadsworth et al. (1986).
If you do not specify the upper and lower limits in the
SPEC statement or the SPEC= data set, then k is assigned
a missing value.
Boyles' Index C_{pm}^{+}
Boyles (1992) proposed
the process capability index C_{pm}^{+}
which is defined as

C_{pm}^{+} = (1/3) [ [( E_{X<T} [ (XT)^{2} ] )/( (T  LSL)^{2} )] + [( E_{X>T} [ (XT)^{2} ] )/( (USL  T)^{2} )] ] ^{1/2}
He proposed this index as a modification of
C_{pm} for use when
.The quantities
and
are referred to as semivariances.
Kotz and Johnson (1993) point out that
if T = (LSL + USL ) / 2,
then
C_{pm}^{+} = C_{pm}.
Kotz and Johnson (1993)
suggest that a natural estimator for C_{pm}^{+} is
Note that this index is not defined when either of the
specification limits is equal to the target T.
Refer to Section 3.5 of
Kotz and Johnson (1993)
for further details.
The Index C_{jkp}
Johnson et al. (1992)
introduced a socalled
"flexible"
process capability index
which takes into account possible differences in variability
above and below the target T.
They
defined this index as
where d = ( USL  LSL ) / 2.
A natural estimator of this index is
For further details, refer to Section 4.4 of Kotz and Johnson (1993).
The Indices C_{pm}(a)
The class of capability indices
C_{pm}(a), indexed by the parameter a
(a>0) allows flexibility
in choosing between the relative importance of
variability and deviation of the mean from
the target value T.
The class defined as
where .The motivation for this definition is that
if is small, then
A natural estimator of C_{pm}(a) is
where
d = ( USL  LSL ) / 2.
You can specify the value of a with the
CPMA= option in
the PROC CAPABILITY statement.
By default, a=0.5.
This index is not recommended for situation in which
the target T is not equal to the
midpoint of the specification limits.
For additional details, refer to Section 3.7 of
Kotz and Johnson (1993).
The Index C_{p(5.15)}
Johnson et al. (1992)
suggest the class of process capability indices
defined as
where is chosen so that the proportion of
conforming items is robust with respect to the
shape of the process distribution.
In particular,
Kotz and Johnson (1993) recommend use of
which is estimated as
For details, refer to Section 4.3.2 of
Kotz and Johnson (1993).
The Index C_{pk(5.15)}
Similarly,
Kotz and Johnson (1993) recommend use of
the robust capability index
where d = ( USL  LSL ) / 2.
This index
is estimated as
For details, refer to Section 4.3.2 of
Kotz and Johnson (1993).
The Index C_{pmk}
Pearn et al. (1992) proposed the index
C_{pmk}
where m = ( LCL + UCL) / 2.
A natural estimator
for
C_{pmk} is
where
m = ( USL + LSL ) / 2.
For further details, refer to Section 3.6 of Kotz and Johnson (1993).
Wright's Index C_{s}
Wright (1995) defines
the capability index
where
.A natural estimator of C_{s} is
where
c_{4} is
an unbiasing constant for the
sample standard deviation,
and
b_{3} is a measure of skewness.
Wright (1995)
shows that
C_{s}
compares favorably with
C_{pmk}
even when skewness is not present,
and he advocates the use of
C_{s}
for monitoring nearnormal processes
when loss of capability typically leads to asymmetry.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.