In a previous article, I showed how a control chart is a major tool in
statistical process control. A control chart with the center line being
the sample average to monitor the process mean, upper control limit and
lower control limit, it's important to know the sample size that will
give the size of the shift we wish to detect in the process. It's also
very important to know the frequency in which we wish to sample.

Suppose we have a process where we are measuring the average ring
diameter over many samples. Let's say the sample average which is the
center line is 64 mm and the probability of detecting a shift from 64 mm
to 64.02 mm increases as the sample size increases. If the process is
large, we use smaller sample sizes than those if the shift of interest
is relatively small. In general, larger samples will make it easier to
detect small shifts in the process.For example, maybe a sample size of
15 will detect a shift of .01 thirty percent of the time. A sample size
of 10 may only detect a shift of .01 twenty percent of the time.

As for the frequency of sampling, the most desirable situation is to
take large sample sizes very frequently to best detect even the smallest
shifts in the process. But that is usually not possible economically.
What is general done is small sample sizes can be taken at short
intervals, say sample of 10 every 15 minutes. Or take larger sample
sizes at larger intervals, say sample of size 50 every hour.

Another way to handle sample size and frequency is through what is known
as the "average run length" of a control chart, or ARL. It's
calculation is simple, just 1 divided by p, where p is the probability
that any point is out of control (outside of the upper control limit or
lower control limit).

For example, if a control chart has limits
of 3 standard deviations from the mean (center line), then ARL =
1/0.0027 = 370. Note that .0027 is found from a cumulative standard
normal distribution chart. Many calculators will calculate this as well.

What does the ARL mean? It means that the average run length of any
process in control is 370. In other words, even if the process is in
control, an out of control signal will occur every 370 samples, on
average.

There are other factors to be taken into account when
trying to more accurately answer the questions of sampling frequency.
Such factors include the cost of sampling, probabilities of various
types of shifts in the process to occur, losses associated with an out
of control process and more.

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