The np-chart with 3-sigma limits and the ARL-unbiased np-chart revisited

Stochastics and Quality Control, 37 (2022), 107-116
https://doi.org/10.1515/eqc-2022-0032

In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters (n,p), where n and p represent the fixed sample size and the fraction nonconforming. In this paper, the traditional np-chart with 3-sigma control limits is reexamined. We show that, even if its lower control limit is positive and we are dealing with a small target value p0 of the fraction nonconforming (p), this chart average run length (ARL) function achieves a maximum to the left of p0. Moreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size n. We also look closely at the ARL function of the ARL-unbiased np-chart proposed by Morais (2016), which attains a pre-specified maximum value in the in-control situation. This chart triggers a signal at sample t with probability one, if the observed number of nonconforming items, xt, is beyond the lower and upper control limits (L and U); probability gL (resp. gU), if xt coincides with L (resp. U). A graphical display for the ARL-unbiased np-chart is proposed, taking advantage of the qcc package for the R statistical software. Furthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.