The Index Cp

The process capability index , sometimes called the "process potential index," the "process capability ratio," or the "inherent capability index," is estimated as

where USL is the upper specification limit, LSL is the lower specification limit, and is the sample standard deviation. If you do not specify both the upper and the lower specification limits in the SPEC statement or the SPEC= data set, then is assigned a missing value.

The interpretation of can depend on the application, on past experience, and on local practice. However, broad guidelines for interpretation have been proposed by several authors. Ekvall and Juran (1974) classify values as

• "not adequate" if

• "adequate" if , but requiring close control as approaches 1

• "more than adequate" if

Montgomery (1996) recommends minimum values of as

• 1.33 for existing processes

• 1.50 for new processes or for existing processes when the variable is critical (for example, related to safety or strength)

• 1.67 for new processes when the variable is critical

Exact lower and upper confidence limits for (denoted by LCL and UCL) are computed using percentiles of the chi-square distribution, as indicated by the following equations:

Here, denotes the lower th percentile of the chi-square distribution with degrees of freedom. Refer to Chou, Owen, and Borrego (1990) and Kushler and Hurley (1992).

You can specify with the ALPHA= option in the PROC CAPABILITY statement or with the CIINDICES( ALPHA=value ) in the PROC CAPABILITY statement. The default value is 0.05. You can save these limits in the OUT= data set by specifying the keywords CPLCL and CPUCL in the OUTPUT statement. In addition, you can display these limits on plots produced by the CAPABILITY procedure by specifying the keywords in the INSET statement.

The Index CPL

The process capability index CPL is estimated as

where is the sample mean, LSL is the lower specification limit, and is the sample standard deviation. If you do not specify the lower specification limit in the SPEC statement or the SPEC= data set, then CPL is assigned a missing value.

Montgomery (1996) refers to CPL as the "process capability ratio" in the case of one-sided lower specifications and recommends minimum values as follows:

• 1.25 for existing processes

• 1.45 for new processes or for existing processes when the variable is critical

• 1.60 for new processes when the variable is critical

Exact lower and upper confidence limits for CPL are computed using a generalization of the method of Chou, Owen, and Borrego (1990), who point out that the lower confidence limit for CPL (denoted by CPLLCL )satisfies the equation

where has a non-central distribution with degrees of freedom and noncentrality parameter . You can specify with the ALPHA= option in the PROC CAPABILITY statement. The default value is 0.05. The confidence limits can be saved in an output data set by specifying the keywords CPLLCL and CPLUCL in the OUTPUT statement. In addition, you can display these limits on plots produced by the CAPABILITY procedure by specifying these keywords in the INSET statement.

The Index CPU

The process capability index CPU is estimated as

where USL is the upper specification limit, is the sample mean, and is the sample standard deviation. If you do not specify the upper specification limit in the SPEC statement or the SPEC= data set, then CPU is assigned a missing value.

Montgomery (1996) refers to CPU as the "process capability ratio" in the case of one-sided upper specifications and recommends minimum values that are the same as those specified previously for CPL.

Exact lower and upper confidence limits for CPU are computed using a generalization of the method of Chou, Owen, and Borrego (1990), who point out that the lower confidence limit for CPU (denoted by CPULCL )satisfies the equation

where has a non-central distribution with degrees of freedom and noncentrality parameter . You can specify with the ALPHA= option in the PROC CAPABILITY statement. The default value is 0.05. The confidence limits can be saved in an output data set by specifying the keywords CPULCL and CPUUCL in the OUTPUT statement. In addition, you can display these limits on plots produced by the CAPABILITY procedure by specifying these keywords in the INSET statement.

The Index Cpk

The process capability index is defined as

Note that the indices , , and are related as . The CAPABILITY procedure estimates as

where USL is the upper specification limit, LSL is the lower specification limit, is the sample mean, and is the sample standard deviation.

If you specify only the upper limit in the SPEC statement or the SPEC= data set, then is computed as CPU, and if you specify only the lower limit in the SPEC statement or the SPEC= data set, then is computed as CPL.

Bissell (1990) derived approximate two-sided 95% confidence limits for by assuming that the distribution of is normal. Using Bissell’s approach, 100% lower and upper confidence limits can be computed as

where denotes the cumulative standard normal distribution function. Kushler and Hurley (1992) concluded that Bissell’s method gives reasonably accurate results.

You can specify with the ALPHA= option in the PROC CAPABILITY statement. The default value is 0.05. These limits can be saved in an output data set by specifying the keywords CPKLCL and CPKUCL in the OUTPUT statement. In addition, you can display these limits on plots produced by the CAPABILITY procedure by specifying these same keywords in the INSET statement.

The Index Cpm

The process capability index is intended to account for deviation from the target in addition to variability from the mean. This index is often defined as

A closely related version of is the index

where and . If , then . However, if , then both indices suffer from problems of interpretation, as pointed out by Kotz and Johnson (1993), and their use should be avoided in this case.

The CAPABILITY procedure computes an estimator of as

where is the sample standard deviation.

If you specify only a single specification limit SL in the SPEC statement or the SPEC= data set, then is estimated as

Boyles (1991) proposed a slightly modified point estimate for computed as

Boyles also suggested approximate two-sided 100% confidence limits for , which are computed as

Here denotes the lower th percentile of the chi-square distribution with degrees of freedom, where equals

You can specify with the ALPHA= option in the PROC CAPABILITY statement. The default value is 0.05. These confidence limits can be saved in an output data set by specifying the keywords CPMLCL and CPMUCL in the OUTPUT statement. In addition, you can display these limits on plots produced by the CAPABILITY procedure by specifying these keywords in the INSET statement.