IRCHART Statement: SHEWHART Procedure

Methods for Estimating the Standard Deviation

When control limits are computed from the input data, three methods (referred to as default, MAD and MMR) are available for estimating the process standard deviation $\sigma $.

Default Method

The default estimate for $\sigma $ is

\[ \hat{\sigma } = \bar{R}/d_{2}(n) \]

where $\bar{R}$ is the average of the moving ranges, n is the number of consecutive individual measurements used to compute each moving range, and the unbiasing factor $d_{2}(n)$ is defined so that if the observations are normally distributed, the expected value of $R_{i}$ is

\[ E(R_{i}) = d_{2}(n_ i)\sigma \]

This method is described in the American Society for Testing and Materials (1976).

MAD Method

If you specify SMETHOD= MAD, a median absolute deviation estimator is computed for $\sigma $, as described by Boyles (1997). It is computed as

\[ \hat{\sigma } = \textrm{median}\{ |X_ i - \tilde{X}|, 1 \leq i \leq N\} /0.6745 \]

where $\tilde{X}$ is the sample median.

MMR Method

If you specify SMETHOD=MMR, a median moving range estimator is computed for $\sigma $. This estimator is described by Boyles (1997). It is computed as

\[ \hat{\sigma } = \tilde{R}/0.954 \]

where $\tilde{R}$ is the median of the nonmissing moving ranges.