13.5Control Charts for Number of Defects 559
and so
UCL=.034+ 3√
(.034)(.968)
50=.1109LCL=.034− 3√
(.034)(.966)
50=−.0429Since the proportion of defectives in the first subgroup falls outside the upper control limit,
we eliminate that subgroup and recomputeFas
F=34 − 6
950=.0295The new upper and lower control limits are .0295±
√
(.0295)(1−.0295)/50, orLCL=−.0423, UCL=.1013Since the remaining subgroups all have fraction defectives that fall within the control limits,
we can accept that, when in control, the fraction of defective items in a subgroup should
be below .1013. ■
REMARK
Note that we are attempting to detect any change in quality even when this change results
in improved quality. That is, we regard the process as being “out of control” even when
the probability of a defective item decreases. The reason for this is that it is important to
notice any change in quality, for either better or worse, to be able to evaluate the reason
for the change. In other words, if an improvement in product quality occurs, then it is
important to analyze the production process to determine the reason for the improvement.
(That is, what are we doing right?)
13.5 Control Charts for Number of Defects
In this section, we consider situations in which the data are the numbers of defects in units
that consist of an item or group of items. For instance, it could be the number of defective
rivets in an airplane wing, or the number of defective computer chips that are produced
daily by a given company. Because it is often the case that there are a large number of
possible things that can be defective, with each of these having a small probability of actually
being defective, it is probably reasonable to assume that the resulting number of defects
has a Poisson distribution.* So let us suppose that, when the process is in control, the
number of defects per unit has a Poisson distribution with meanλ.
* See Section 5.2 for a theoretical explanation.