vdoc

256 Part IV: Quality Assurance
selection of a subgroup or sample is perhaps the most important item in setting up
a control chart. The next question in the selection of a sample is to determine the
sample size. Factors that are usually taken into consideration for determining
the sample size and the frequency of the samples are the average run length (ARL)
and the operating characteristic curve, which is also known as the OC curve.
AVERAGE RUN LENGTH
A run is a number of successive items possessing the same characteristics. For
example, number of successive conforming or successive nonconforming forms a
run. An average run length (ARL) is the average number of points plotted, that is,
the number of subgroups inspected before a point falls outside the control limits,
indicating that the process is out of control.
In Shewhart control charts the ARL can be determined by using the formula
ARL = 1 p
(19.1)
where p is the probability that any point falls outside the control limits. It is quite
common to use ARL as a benchmark to check the performance of a control chart.
As an illustration, consider a process quality characteristic that is normally
distributed. Then for an X – control chart with three-sigma control limits, the probability
that a point will fall outside the control limits when the process is stable is
p = 0.0027, which is the probability that a normal random variable deviates from
the mean m by at least three s. Thus, the average run length for the X – control chart
when the process is stable is
Part IV.B
1
ARL 0
= =370.
0.
0027
In other words, when the process is stable we should expect that, on the average,
one out-of-control signal or false alarm will occur once in every 370 samples. The
ARL can also be used to determine how often a false alarm will occur, simply by
multiplying the ARL 0 by the time t between the samples. Thus, for example, if
samples are taken every 30 minutes, then a false alarm will occur on the average
once every 185 hours. On the other hand, ARL can be used in the same manner to
find how long it will take before a given shift in the process mean is detected. We
illustrate this concept with Example 19.2 using an X – control chart.
In practice, the decision of how large and how frequently the samples should
be taken is based on the cost of taking samples and how quickly we would like to
detect the shift. Large samples taken more frequently would certainly give better
protection against shifts, since it will take less time to detect any given shift.
Thus, for instance, in Example 19.2, if the samples are taken every half an hour
instead of every hour then it will take only one hour instead of taking two hours
to detect the shift of 1.5s. Similarly, it can easily be shown that if larger samples
are taken, the shifts can be detected more quickly. This means if large samples are
taken more frequently, shifts in the process mean will be detected faster, and the
process will be producing fewer nonconforming units. Thus, when calculating