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where ''Q''<sub>''i''</sub>(''n''<sub>''i''</sub>,'''''X''<sub>''i''</sub>''') denotes a statistical decision rule, ''n<sub>i</sub>'' denotes the size of the sample '''S'''<sub>''i''</sub>, that is the number of the samples the rule is applied upon, and '''X'''<sub>''i''</sub> denotes the vector of the rule specific parameters, including the decision limits. Each symbol ''#'' denotes either the [[Boolean logic|Boolean]] operator AND or the operator OR. Obviously, for ''#'' denoting AND, and for ''n''<sub>1</sub> < ''n''<sub>2</sub> <...< ''n''<sub>''q''</sub>, that is for '''S'''<sub>1</sub> <math>\subset</math> '''S'''<sub>2</sub> <math>\subset</math> ....<math>\subset</math> '''S'''<sub>''q''</sub>, the (1) denotes a ''q''-sampling QC procedure.
Each statistical decision rule is evaluated by calculating the respective statistic of a monitored variable of samples taken from the process. Then, if the statistic is out of the interval between the decision limits, the decision rule is considered to be true. Many statistics can be used, including the following: a single value of the variable of a sample, the [[range (statistics)|range]], the [[mean]], and the [[standard deviation]] of the values of the variable of the samples, the cumulative sum, the smoothed mean, and the smoothed standard deviation. Finally, the QC procedure is evaluated as a Boolean proposition. If it is true, then the [[null hypothesis]] is considered to be false, the process is considered to be out of control, and the run is rejected.
A [[quality control]] procedure is considered to be optimum when it minimizes (or maximizes) a context specific objective function. The objective function depends on the probabilities of detection of the nonconformity of the process and of false rejection. These probabilities depend on the parameters of the [[quality control]] procedure (1) and on the probability density functions (see [[probability density function]]) of the monitored variables of the process.
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