XCHART Statement: CUSUM Procedure

Basic Notation for Cusum Charts

The following notation is used in this chapter:

$\mu$	denotes the mean of the population, also referred to as the process mean or the process level.
$\mu _{0}$	denotes the target mean (goal) for the population. Goel and Wu (1971) refer to $\mu _{0}$ as the “acceptable quality level” and use the symbol $\mu _{a}$ instead. The symbol $\bar{X}_{0}$ is used for $\mu _{0}$ in Glossary and Tables for Statistical Quality Control. You can provide $\mu _{0}$ with the MU0= option or with the variable `_MU0_` in a LIMITS= data set.
$\sigma$	denotes the population standard deviation. You can provide $\sigma$ with the variable `_STDDEV_` in a LIMITS= data set (where `_TYPE_`='STANDARD').
$\sigma _{0}$	denotes a known standard deviation. You can provide $\sigma _{0}$ with the SIGMA0= option or the variable `_STDDEV_` in a LIMITS= data set.
$\hat{\sigma }$	denotes an estimate of $\sigma$ . You can provide $\hat{\sigma }$ with the SIGMA0= option or the variable `_STDDEV_` in a LIMITS= data set. To identify this value as an estimate, specify TYPE=ESTIMATE or assign the value 'ESTIMATE' to the variable `_TYPE_` in a LIMITS= data set.
n	denotes the nominal sample size for the cusum scheme. You can provide n with the LIMITN= option or the variable `_LIMITN_` in a LIMITS= data set.
$\delta$	denotes the shift in $\mu$ to be detected, expressed as a multiple of the standard deviation. You can provide $\delta$ with the DELTA= option or the variable `_DELTA_` in a LIMITS= data set.
$\Delta$	denotes the shift in $\mu$ to be detected, expressed in data units. If the sample size n is constant across subgroups, then $\Delta =\delta \sigma _{\bar{X}}= (\delta \sigma )/\sqrt {n}$ .
	Some authors use the symbol D instead of $\Delta$ ; for example, refer to Johnson and Leone (1962, 1974) and Wadsworth, Stephens, and Godfrey (1986). You can provide $\Delta$ with the SHIFT= option. Although it may be more natural to specify the shift in data units, it is preferable to specify the shift as $\delta$ , since this generalizes to data with unequal subgroup sample sizes.