Q. Which is the better chart to use
for analysis of yield data: the p-chart or the
individual-moving range chart? I am trying to
better understand the changes in the weekly
test yields on our electronic product lines.
I've used both p-charts and individual charts
with different results. The p-chart seems much
more sensitive and provides many more out-of-control
indications than the individual chart.
A. Your observation that the p-chart
is more sensitive is correct and statistically
expected. The p-chart is designed for data in
the form of proportions and takes into account
the sample size when calculating the control
limits. As the sample size goes up, the sensitivity
of all control charts increases. This increased
sensitivity means that the p-chart is better
able to detect process changes as they occur.
Although it is technically possible to make
an individual chart using the individual yield
proportions, it is not the correct chart to
use for this type of data, as explained below.
Individual Chart Concerns
There are four primary problems with using
an individual chart for yield data:
(1) The estimate of the sample standard deviation
may be larger than is correct because the individual
chart assumes that each proportion is based
on only one observation, rather than larger
size samples. Recall that the standard deviation
of a sample decreases as the sample size increases
because the sample standard deviation is calculated
as 
.
In addition, the individual chart usually calculates
the standard deviation based on a moving range
of size 2, rather than looking at all of the
data. This method can be affected by trends
and "noise" in the data.
(2) If the sample sizes vary between observations,
the calculated center line may be different
than the average proportion based on all of
the samples. This is because the individual
chart considers each observed proportion equally,
whether it is based on 2 samples or 2000 samples.
(3) An individual chart has constant control
limits, rather than control limits based on
the size of each sample, ni.
(4) Serious departures from normality can have
an adverse effect on the individual chart, while
the p-chart should work fairly well as long
as np > 5 and n(1-p)
> 5.
P-Chart is Specifically Designed for Proportions
The p-chart is more sensitive and more correct
in the case of yield data since it takes the
sample size into account when calculating the
control limits. The p-chart was specifically
designed to use data that is the proportion
of observations meeting some criteria out of
each sample of size ni. The
sample size ni goes into calculating
the control limits as:
where 
is the average proportion rejected based on
all of the samples. Note that larger sample
sizes result in narrower, more sensitive, control
limits. This chart also has variable width control
limits if the sample sizes are not all the same.
Example
The following data represents the number of
defects, the sample size, and the proportion
rejected for each sample.
|
Sample
|
Defects
|
Sample Size (ni)
|
Proportion Rejected
|
|
1
|
12
|
500
|
0.024
|
|
2
|
7
|
300
|
0.023
|
|
3
|
10
|
500
|
0.020
|
|
4
|
11
|
200
|
0.055
|
|
5
|
22
|
600
|
0.037
|
|
6
|
13
|
550
|
0.024
|
|
7
|
10
|
400
|
0.025
|
|
8
|
52
|
1000
|
0.052
|
|
9
|
9
|
500
|
0.018
|
|
10
|
8
|
450
|
0.018
|
The incorrect application of an individual
chart is shown first. Note that the upper control
limit is quite high (0.69) and the lower control
limit is less than zero (-0.01). However, the
true proportion rejected cannot be less than
zero. This individual chart does not indicate
any out-of-control points.
The correct application of a p-chart is shown
below. Note that the control limits have now
been adjusted to account for the various sample
sizes encountered. Larger samples correctly
have narrower control limits, and all of the
control limits are narrower than calculated
for the individual chart. The lower control
limits are not less than zero. Point 8, calculated
on a sample size of 1000, is clearly out of
control based on this p-chart. However, point
4, with a similar proportion rejected but with
a sample size of only 200, is not out of control.
