Basic Notation for Cusum Charts

XCHART Statement

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 and others (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.

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$\mu$	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 and others (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.