SPCAPABILITY procedure

Calculates capability statistics (R.W. Payne).

Option

PRINT = strings

Controls output (cpk, ppk, histogram); default cpk, ppk

Parameters

DATA = variates or pointers

Data measurements

SAMPLES = factors or scalars

Factor identifying samples or scalar indicating the size of each sample

LOWERLIMIT = scalars

Specifies the lower specification limit for each set of data

UPPERLIMIT = scalars

Specifies the upper specification limit for each set of data

CPK = scalars

Saves the index C_pk

PPK = scalars

Saves the index P_pk

Description

SPCAPABILITY calculates capability statistics. These are used to assess the extent to which the output of a process lies within its specification limits. The data values consist of samples of measurements made on successive occasions, which are specified by the DATA and SAMPLES parameters. DATA can be set to a variate containing the measurement and SAMPLES to a factor identifying the samples. Alternatively, if the samples are all of the same size and occur in the DATA variate one sample at a time, you can set SAMPLES to a scalar indicating the size of each sample. Finally, if the samples are in separate variates, you can set DATA to a pointer containing the variates (SAMPLES is then unset). The LOWERLIMIT parameter supplies the lower specification limit of the process, and the UPPERLIMIT parameter supplies the upper limit.

There are two indexes that can be calculated. The index C_pk is the minimum of the two quantities C_pl and C_pu. These are defined as

C_pl = (LOWERLIMIT - mean) / (3 × sigma)

C_pu = (UPPERLIMIT - mean) / (3 × sigma)

where sigma is the within-sample standard deviation (see for example Ryan 1989, Chapter 7). The alternative index, P_pk, is the minimum of the two quantities P_pl and P_pu. These have similar definitions to C_pl and C_pu, except that sigma now also includes the between-sample variation.

The PRINT option controls which of these are printed, with settings cpk and ppk. There is also a setting histogram, which plots a histogram of the data together with vertical lines indicating the lower and upper limits. By default PRINT=cpk,ppk. Alternatively, the indexes can be saved, in scalars, using the parameters CPK and PPK.

Option: PRINT.

Parameters: DATA, SAMPLES, LOWERLIMIT, UPPERLIMIT, CPK, PPK.

Method

SPCAPABILITY estimates the within-sample standard deviation (for C_pk) by the average of the standard deviations of the samples, each divided by a bias correction constant c₄:

c₄ = √(2/n) × GAMMA(n/2) / GAMMA((n-1)/2)

where n is the sample size. Similarly the standard deviation for P_pk is the bias-corrected standard deviation of the samples, all pooled together.

Action with RESTRICT

Neither the DATA variates nor the SAMPLE factors may be restricted.

Reference

Ryan, T.P. (1989). Statistical Methods for Quality Improvement. Wiley, New York.