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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

\[ \widehat{C}_ p = \frac{\mi {USL} - \mi {LSL}}{6s} \]

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:

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

Here, $\chi ^{2}_{\alpha ,\nu }$ denotes the lower $100\alpha $th percentile of the chi-square distribution with $\nu $ degrees of freedom. Refer to Chou, Owen, and Borrego (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

\[ \widehat{ \mr {CPL} } = \frac{\bar{X} - \mi {LSL}}{3s} \]

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, Owen, and Borrego (1990), who point out that the $100(1-\alpha )$ lower confidence limit for CPL (denoted by CPLLCL )satisfies the equation

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

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

\[ \widehat{ \mr {CPU} } = \frac{\mi {USL} - \bar{X}}{3s} \]

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, Owen, and Borrego (1990), who point out that the $100(1-\alpha )$ lower confidence limit for CPU (denoted by CPULCL )satisfies the equation

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

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

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

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

\[ \widehat{C}_{pk} = \frac{1}{3s} \times \min (\mi {USL} - \bar{X},~ \bar{X} - \mi {LSL}) = \min (\mi {CPU},~ \mi {CPL}) \]

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 $\widehat{C}_{pk}$ is normal. Using Bissell’s approach, 100$(1-\alpha )$% lower and upper confidence limits can be computed as

$\displaystyle \mbox{lower limit} $
$\displaystyle = $
$\displaystyle \widehat{C}_{pk} \left[ 1 - \Phi ^{-1}(1-\alpha /2 ) \sqrt { \frac{1}{9n \widehat{C}_{pk}^{2} } + \frac{1}{2(n-1)} } \; \right] $
$\displaystyle \mbox{upper limit} $
$\displaystyle = $
$\displaystyle \widehat{C}_{pk} \left[ 1 + \Phi ^{-1}( 1-\alpha /2 ) \sqrt { \frac{1}{9n \widehat{C}_{pk}^{2} } + \frac{1}{2(n-1)} } \; \right] $

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

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

A closely related version of $C_{pm}$ is the index

\[ C_{pm}^{*} = \frac{ \min \left( \mr {USL} - T, T - \mr {LSL} \right) }{ 3 \sqrt { \sigma ^2 + ( \mu - T )^2 } } = \frac{ d - |T - m| }{ 3 \sqrt { \sigma ^2 + ( \mu - T )^2 } } \]

where $d = ( \mr {USL} - \mr {LSL} ) / 2$ and $m = ( \mr {USL} + \mr {LSL} ) / 2$. 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

\[ \widehat{C}_{pm} = \frac{\min (\mi {USL} - T,~ T - \mi {LSL})}{3 \sqrt {s^2 + (\bar{X} - T)^2}} \]

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

\[ \widehat{C}_{pm} = \frac{|T - \mi {SL}|}{3 \sqrt {s^2 + (\bar{X} - T)^2}} \]

Boyles (1991) proposed a slightly modified point estimate for $C_{pm}$ computed as

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

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

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

Here $\chi ^{2}_{\alpha ,\nu }$ denotes the lower $100\alpha $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.

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