ABSTRACT
The central line and warning and action limits may be derived from studying pilot samples taken when the process is presumed or known to be in control, or alternatively may be fixed by a priori considerations. If they are derived from pilot samples we shall assume
that they are of the same size as those to be taken when the control scheme is in operation and that the mean and range R are calculated for each such sample. These quantities are then averaged over all the pilot samples to obtain and . We may also calculate, instead of R, either the unadjusted or the adjusted sample standard deviations S or s (see below). The charts are then drawn up as follows:
As an alternative to using pilot samples, specifications of the mean µ and/or the standard deviation σ of the process measurements may be used to define the ‘in control’ situation. If µ is given, use it in place of in drawing up the . If σ is given, the expected value of R is equal to d1σ, so here define as d1σ and then proceed as above. This application allows an exact interpretation to be made of the warning and action limits, for if the process measurements are normally distributed with mean µ and standard deviation σ the warning limits thus obtained correspond to quantiles q of 0.025 and 0.975 and the action limits to quantiles of 0.001 and 0.999. In other words, the limits can be regarded as critical values for testing the null hypothesis that the data are indeed from a normal distribution with mean µ and standard deviation σ, the warning limits corresponding to significance levels of or α2=5% and the action limits to levels of α1=0.1% or α2=0.2%.
