Standard Capability Indices

PROC CAPABILITY and General Statements

Standard Capability Indices

This section provides computational details for the standard process capability indices computed by the CAPABILITY procedure: C_p, CPL, CPU, C_pk, and C_pm.

The Index Cp

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

$\hat{C}_p = \frac{{USL} - {LSL}}{6s}$

where USL is the upper specification limit, LSL is the lower specification limit, and s 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 C_p is assigned a missing value.

The interpretation of C_p 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 C_p values as

"not adequate" if C_p < 1
"adequate" if $1 \leq C_p \leq 1.33$ , but requiring close control as C_p approaches 1
"more than adequate" if C_p > 1.33

Montgomery (1996) recommends minimum values of C_p 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 $100(1-\alpha)\%$ lower and upper confidence limits for C_p (denoted by LCL and UCL) are computed using percentiles of the chi-square distribution, as indicated by the following equations:

${lower limit} & = & \hat{C_{p}} \sqrt{ \chi^2_{\alpha/2,n-1} / (n-1) } \{upper limit} & = & \hat{C_{p}} \sqrt{ \chi^2_{1-\alpha/2,n-1} / (n-1) }$

Here, $\chi^2_{\alpha,\nu}$ denotes the lower $100\alpha\!^{{\scriptsize th}}$ percentile of the chi-square distribution with $\nu$ degrees of freedom. Refer to Chou et al. (1990) and Kushler and Hurley (1992).

You can specify $\alpha$ 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

$\hat{ {\rm CPL} } = \frac{\bar{X} - {LSL}}{3s}$

where $\bar{X}$ is the sample mean, LSL is the lower specification limit, and s 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 $100(1-\alpha)\%$ lower and upper confidence limits for CPL are computed using a generalization of the method of Chou et al. (1990), who point out that the $100(1-\alpha)$ lower confidence limit for CPL (denoted by CPLLCL )satisfies the equation

${Pr}\{ T_{n-1}(\delta = 3\sqrt{n}) { CPLLCL } \leq 3 {CPL} \sqrt{n} \} = 1 - \alpha$

where $T_{n-1}(\delta)$ has a non-central t distribution with n-1 degrees of freedom and noncentrality parameter $\delta$ .You can specify $\alpha$ 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

$\hat{ {\rm CPU} } = \frac{{USL} - \bar{X}}{3s}$

where USL is the upper specification limit, $\bar{X}$ is the sample mean, and s 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 $100(1-\alpha)\%$ lower and upper confidence limits for CPU are computed using a generalization of the method of Chou et al. (1990), who point out that the $100(1-\alpha)$ lower confidence limit for CPU (denoted by CPULCL )satisfies the equation

${Pr}\{ T_{n-1}(\delta = 3\sqrt{n} { CPULCL } \geq 3 {CPU} \sqrt{n} \} = 1 - \alpha$

where $T_{n-1}(\delta)$ has a non-central t distribution with n-1 degrees of freedom and noncentrality parameter $\delta$ .You can specify $\alpha$ 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 C_pk is defined as

$C_{pk} = \frac{1}{3 \sigma} \min({USL} - \mu,\mu - {LSL}) = \min({CPU},{CPL})$

Note that the indices C_pk, C_p, and k are related as C_pk = C_p (1 - k). The CAPABILITY procedure estimates C_pk as

$\hat{C}_{pk} = \frac{1}{3s} x \min({USL} - \bar{X},\bar{X} - {LSL}) = \min({CPU},{CPL})$

where USL is the upper specification limit, LSL is the lower specification limit, $\bar{X}$ is the sample mean, and s is the sample standard deviation.

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

Bissell (1990) derived approximate two-sided 95% confidence limits for C_pk by assuming that the distribution of $\hat{C}_{pk}$ is normal. Using Bissell's approach, 100 $(1-\alpha)$ % lower and upper confidence limits ycan be computed as

${lower limit} & = & \hat{C}_{pk} [ 1 - \Phi^{-1}(1-\alpha/2 ) \sqrt{ \frac{1}{9n... ...-1}( 1-\alpha/2 ) \sqrt{ \frac{1}{9n \hat{C}_{pk}^2 } + \frac{1}{2(n-1)} } \; ]\$

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

You can specify $\alpha$ 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 C_pm is intended to account for deviation from the target T in addition to variability from the mean. This index is often defined as

$C_{pm} = \frac{\rm{USL} - \rm{LSL}} { 6 \sqrt{\sigma^2 + (\mu - T)^2 } }$

A closely related version of C_pm is the index

$C_{pm}^{*} = \frac{ \min ( \rm{USL} - T, T - \rm{LSL} ) } { 3 \sqrt{ \sigma^2 ... ...u - T )^2 } } = \frac{ d - | T - m| } { 3 \sqrt{ \sigma^2 + ( \mu - T )^2 } }$

where d = ( USL - LSL ) / 2 and m = ( USL + LSL ) / 2. If T=m, then C_pm = C_pm^*. However, if $T \neq m$ ,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 C_pm as

$\hat{C}_{pm} = \frac{\min({USL} - T,T - {LSL})} {3 \sqrt{s^2 + (\bar{X} - T)^2}}$

where s is the sample standard deviation.

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

$\hat{C}_{pm} = \frac{| T - {SL}|}{3 \sqrt{s^2 + (\bar{X} - T)^2}}$

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

$\widetilde{C}_{pm} = \frac{({USL} - {LSL})/2} {3 \sqrt{(\frac{n-1}n)s^2 + (\bar{X} - T)^2}}$

Boyles also suggested approximate two-sided 100 $(1-\alpha)$ % confidence limits for C_pm, which are computed as

${lower limit} & = & \widetilde{C}_{pm} \sqrt{ \chi^2_{\alpha/2,\nu} / \nu }\{upper limit} & = & \widetilde{C}_{pm} \sqrt{ \chi^2_{1-\alpha/2,\nu} / \nu }\$

Here $\chi^2_{\alpha,\nu}$ denotes the lower $100\alpha\!^{{\scriptsize th}}$ percentile of the chi-square distribution with $\nu$ degrees of freedom, where $\nu$ equals

$\frac{n(1+(\frac{\bar{X} - T}s)^2)} {1+2(\frac{\bar{X} - T}s)^2}$

You can specify $\alpha$ 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.

Chapter Contents
Previous
Next
Top