XRCHART Statement: SHEWHART Procedure

Capability Indices

This section provides formulas for process capability indices, which are saved in the OUTLIMITS= data set when you use the LSL= and USL= options to provide lower and upper specification limits (LSL and USL, respectively) for the process. The estimate $\hat{\sigma }$ is computed as described in the previous section, Methods for Estimating the Standard Deviation

The Index $C_ p$

The process capability index $C_{p}$ is computed as

\[ C_{p} = (USL- LSL)/ 6\hat{\sigma } \]

If you do not specify both LSL and USL, the variable _CP_ is assigned a missing value.

The Index CPL

The process capability index CPL is computed as

\[ CPL = (\overline{\overline{X}} - LSL)/3\hat{\sigma } \]

If you do not specify LSL, the variable _CPL_ is assigned a missing value.

The Index CPU

The process capability index CPU is computed as

\[ CPU = (USL- \overline{\overline{X}})/3\hat{\sigma } \]

If you do not specify USL, the variable _CPU_ is assigned a missing value.

The Index $C_{pk}$

The process capability index $C_{ pk}$ is computed as

\[ C_{pk} = \min (USL- \overline{\overline{X} }, \overline{\overline{X}} - LSL)/ 3\hat{\sigma } \]

If you specify only USL, the index $C_{pk}$ is computed as

\[ C_{pk} = (USL- \overline{\overline{X}} ) / 3\hat{\sigma } \]

and if you specify only LSL, the index $C_{pk}$ is computed as

\[ C_{pk} = (\overline{\overline{X}} - LSL) / 3\hat{\sigma } \]
The Index $C_{pm}$

The process capability index $C_{pm}$ is computed as

\[ C_{pm} = \frac{\min (T-LSL,USL-T)}{3\sqrt {\hat{\sigma }^{2} +(\overline{\overline{X}}-T)^{2}}} \]

where T is the target value specified with the TARGET= option.

When a single specification limit (SL) and target are specified, $C_{pm}$ is computed as

\[ C_{pm} = \frac{|T - SL|}{3\sqrt {\hat{\sigma }^{2} +(\overline{\overline{X}}-T)^{2}}} \]

You can also use the CAPABILITY procedure to compute a variety of capability indices. The SHEWHART procedure and the CAPABILITY procedure use the same formulas to calculate the indices, but they use different estimates for the process standard deviation $\sigma $.

  • The SHEWHART procedure calculates $\hat{\sigma }$ from subgroup estimates of $\sigma $. For details, see the previous section, "Methods for Estimating the Standard Deviation."

  • The CAPABILITY procedure calculates $\hat{\sigma }$ as the sample standard deviation of the entire sample. For details, see the section Standard Deviation.

Regardless of which method you use, you should verify that the process is in statistical control before interpreting the indices, and you should verify that the data are normally distributed. The CAPABILITY procedure provides a variety of statistical and graphical tests for checking normality.

Some references use different notation and names for capability indices. For example, the manual ASQC Automotive Division/AIAG (1990) uses the term "process capability indices" for the indices listed in this section, and it uses the term "process performance indices" for the indices computed by the CAPABILITY procedure.