Q: I am interested in getting your recommendations
concerning using Statit p-chart subroutines
in situations where either the sample size or
the number of defects is small. At what point
does the normal approximation begin to break
down?
A: Small sample (subgroup) sizes (n)
combined with a small proportion defective (p)
can hurt the ability of the p-chart to detect
process changes. The p-chart relies on the normal
approximation to the binomial distribution,
which is typically good only if np >
5 and n[1-p] > 5. Design of well-functioning
p-charts is usually assured through the correct
choice of the sample size. There are three primary
approaches to selecting the sample size: which
of the approaches to use depends on the situation
in which the control chart will be used.
Method 1
The sample size can be chosen so that the probability
of finding at least one defect in any sample
(γ) is fairly high (often 90 to 95%).
Otherwise, the presence of only one nonconforming
item in the sample would indicate an out-of-control
situation even though some samples with defects
would be expected to occur since p >
0. This method is used to reduce the number
of "false alarm" indications from
the control chart that do not indicate actual
changes in the process.
Let γ = probability of finding at least
one defect and p = estimated proportion
defective in the population. Then using the
Poisson approximation to the binomial distribution,
the minimum sample size would be:
For example, suppose that the estimated proportion
defective in the population is p = 0.04,
and that we want to have a 90% probability of
finding at least one defective in the sample
(γ = 0.90). Then:
Method 2
A second method of selecting a sample size
is based on the size of the change in the proportion
defective (δ) that you wish to be able
detect. Generally, a 50% chance of detecting
on any one sample is thought to be sufficient,
since if the shift is not detected on the first
sample after the shift, it could be caught on
subsequent samples. Let δ = magnitude
of the shift you want to detect and p
= estimated proportion defective, as above.
Then for the standard k-sigma control
limits, n would be calculated as:
For example, assume again that the proportion
defective is p = 0.04. We would like
to be able to detect if the process shifts up
to 0.10. Thus, the shift we want to detect is
δ = 0.10 – 0.04 = 0.06. With standard
3-sigma control limits, we would calculate the
sample size as:
As expected, the sample size will go up if
smaller shifts are to be detected.
Method 3
The third method is to select the sample size
that will make the lower control limit at least
equal to zero. This method is best used when
the p-chart is being used to determine either
(1) when improvements are being made that decrease
the proportion defective, or (2) when there
may be a problem in the monitoring/reporting
system. Assuming k-sigma control limits, the
sample size would be at least:
For example, for population proportion defective
p = 0.04 and standard 3-sigma control
limits, the sample size to ensure a positive
lower control limit would be:
Notes:
| 1. |
These methods will not recommend
the same sample size, since they are each
looking at different criteria. It is important
to examine the purposes behind the p-chart
for each particular situation. |
| 2. |
If the sample size is too
large to be practical, you may need to combine
subgroups. However, make sure that it makes
sense to do so. All of the observations
in a sample should be logically related,
and the chart must still be able to respond
to process changes in a timely fashion.
|
| 3. |
When defect levels become
very low, there may be very long periods
of time between the occurrences of nonconforming
products. The time (or count of units) between
nonconforming products could be charted.
We’ll deal with this method in another
installment of the Quality Query. |
| 4. |
For more information on these
methods, a good reference is Montgomery,
Douglas, Introduction to Statistical Quality
Control, 3rd Ed., Wiley, 1996. |
