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 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 Cpm+

Boyles (1992) proposed the process capability index Cpm+ which is defined as
Cpm+ = (1/3) [ [( EX<T [ (X-T)2 ] )/( (T - LSL)2 )] + [( EX>T [ (X-T)2 ] )/( (USL - T)2 )] ] -1/2
He proposed this index as a modification of Cpm for use when .The quantities
and
are referred to as semivariances. Kotz and Johnson (1993) point out that if T = (LSL + USL ) / 2, then Cpm+ = Cpm.

Kotz and Johnson (1993) suggest that a natural estimator for Cpm+ 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 Cjkp

Johnson et al. (1992) introduced a so-called "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 Cpm(a)

The class of capability indices Cpm(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 Cpm(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 Cp(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 Cpk(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 Cpmk

Pearn et al. (1992) proposed the index Cpmk
where m = ( LCL + UCL) / 2. A natural estimator for Cpmk is

where m = ( USL + LSL ) / 2.

For further details, refer to Section 3.6 of Kotz and Johnson (1993).

### Wright's Index Cs

Wright (1995) defines the capability index
where .

A natural estimator of Cs is

where c4 is an unbiasing constant for the sample standard deviation, and b3 is a measure of skewness. Wright (1995) shows that Cs compares favorably with Cpmk even when skewness is not present, and he advocates the use of Cs for monitoring near-normal processes when loss of capability typically leads to asymmetry.

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