In Manufacturing, much of the data is attribute
data; that is, counts of things, events or outcomes.
We know that measurement data gives us much
more information, but attribute data often gives
us a good feel for the process as well. Our
counts could be how many misplaced components
are on a PCB, how many parts had mechanical
failures, or how many battery separators with
die lines were packed. Often, the data can be
represented on an i chart. The i chart is almost
always available, but it is seldom the best
choice. Let’s take a look at the charts
that may be better and why.
While the i chart was designed to analyze measurements
in a slow-moving process, one can plot either
a count of events or a rate or proportion. However,
this application of the i chart is likely to
hide important information. For example, if
the i chart is a plot of the number of PCB defects
in a day, it would in no way indicate the number
of boards produced in the period. This is critical
information in deciding whether a spike in the
number is statistically important. The counts
of 1 defect vs. 10 defects may take on a different
meaning if we know that in the former, there
were only two boards inspected while in the
latter, there were 1000 boards inspected.
In the same way, plotting the rates of defects
(defects / total boards) on an i chart does
not take into account the size of the subgroup
in determining the control limits. Some critical
information is hidden from the decision maker.
Statistically, in estimating our confidence
in a rate of .25, it makes a difference if that
rate is 250 events in 1000 trials or 1 in 4.
While there is some information of the subgroup
size in the rate, the i chart does not take
different sizes into account in calculating
the control limits.
A c or u chart or a p chart are usually better
choices for representing these data. The c and
u charts work with what is known as an “Area
of Opportunity” in which any number of
events can occur. The Area of Opportunity is
some unit that presents an opportunity for the
event to occur and is also commonly referred
to as "Subgroup Size". A PCB presents
opportunities for misplaced components. A box
of battery separators presents opportunities
for bad separators being packed. A part presents
an opportunity for a mechanical failure. Often
the choice of a c or u chart over a p chart
is determined by the question, can an event
happen more than once in the area of opportunity?
Can there be more than one misplaced component
on a PCB? Can there be more than one mechanical
failure on a part? Can a box of battery separators
contain more than one separator with die lines.
If the answer is yes, then your data indicate
a c or u chart. These charts allow for those
situations where it is possible to have more
events than areas of opportunity. If your area
of opportunity is a PCB, it is possible to have
more than 1 misplaced component on the board,
since there are likely several components on
the board.
Once you have determined that a c or u chart
is applicable, the decision of which one to
use is dependent on whether the area of opportunity
data are available. A c chart simply plots counts
and assumes a constant size for the area of
opportunity or subgroup. A u chart needs the
size of the area of opportunity for its calculations
because it plots counts per area of opportunity.
If you choose the number of boxes of separators
as the area of opportunity for die lines, then
the c chart could be applicable only if the
number of boxes inspected was fairly constant
across all periods charted. For some indicators,
you may not have the actual size of the area
of opportunity, but can make an educated assumption
that it is fairly constant. This may be dangerous,
however, without data to back up your assumption.
If the area of opportunity varies and if you
have the size of the area of opportunity for
each count, the u chart is the better choice.
The u chart will actually contain more information
than the c chart (or the i chart). As Hart and
Hart (2002) says, “Note that the u chart
may always be used and that the c chart is never
better.”
In u chart data, there are a number of areas
of opportunities in each time period. The u
chart calculates the plotted point by dividing
the number of events by this number of the areas
of opportunities in that time period. Be sure
to note that the number of areas of opportunities
can be fractional.
The area of opportunity for a u chart can be
manipulated to a unit that may be more acceptable
or reasonable to the user community, such as
misplaced components per 1000 PCB or mechanical
failures per 100 parts. This is reasonable since
these charts are based on the Poisson distribution
which is applicable to infrequent events. The
u chart divides the number of events by the
number of units of areas of opportunities. A
chart of the transformed data is what Grant
and Leavenworth (1996) calls the ku chart, simply
because scaling factor, k, is applied to the
count of areas of opportunity, which has the
effect of dividing the number of Areas of Opportunities
by k. Statistically, this scalar manipulation
has no affect on the analytical results but
may scale the numbers to make more sense to
the user. Thus, 6 misplaced components in 290
parts becomes 6 misplaced components in 2.90
100 Parts or .021 misplaced components per 100
parts. This is a simple transformation of dividing
the number of Parts for each period by 100.
In this example, the Area of Opportunity is
Parts and the Unit of Area of Opportunity is
100 Parts.
The p chart is used when there are only two
possible outcomes to some event and the sum
of the counts of the two outcomes equal the
total area of opportunity (the total number
of events). The former indicates a binomial
distribution. The latter says that the count
of outcomes is a subset of the area of opportunity.
Several data in the industry fall into this
group. The part had a mechanical failure or
it did not. If there is a specification on the
number separators with die lines in a box, then
the box either met this specification or did
not.
The area of opportunity for this chart may
actually be a recorded event such as parts produced
or number of boxes of separators . In this case,
the count of an outcome (numerator) is a subset
of the count of events (denominator). The count
of events is the Area of Opportunity. For all
parts produced, how many had mechanical failures?
The event is a part that was produced and the
two possible outcomes are that it had a mechanical
failure or that it did not. The number of parts
with mechanical failures is a subset of the
total parts produced. With this type of data,
the number of a particular outcome divided by
the number of events can never exceed 1.
In contrast, Misplaced Component is not a subset
of PCBs. A number of Components were misplaced,
but no PCB was misplaced.
Another concept to help decide on a u chart
or p chart is whether the actual number of opportunities
for an event can be counted. If counting the
opportunities is very difficult or impossible,
it is necessary to use a u chart instead of
a p chart. If we are counting any type of a
defect in a complex PCB board, it may be difficult
to count all of the opportunities. A classic
example often used is a roll of cloth. Obviously,
the number of opportunities for a thread defect
or color defect would be very hard to count.
This leads one to attempt to find an area of
opportunity that would be more manageable, but
which will have an indicative relationship to
the possibility of an occurrence of the event.
A PCB board for Misplaced Components works because
the PCB contains an indication of the opportunity
for Misplaced Components. A bolt of cloth works
for thread defect for the same reason.
Let’s look at some actual data to compare
these charts.
Misplaced Components
Here are the data for Misplaced Components
for a particular board. The Rate is calculated
as Misplaced/PCB. PCB_100 is PCB/100.
| Date |
Misplaced |
PCB |
PCB_100 |
rate |
| 01-Dec-2005 |
13 |
600 |
6.0 |
0.022 |
| 02-Dec-2005 |
12 |
430 |
4.3 |
0.028 |
| 03-Dec-2005 |
7 |
290 |
2.9 |
0.024 |
| 04-Dec-2005 |
19 |
550 |
5.5 |
0.035 |
| 05-Dec-2005 |
14 |
440 |
4.4 |
0.032 |
| 06-Dec-2005 |
9 |
200 |
2.0 |
0.045 |
| 07-Dec-2005 |
18 |
550 |
5.5 |
0.033 |
| 08-Dec-2005 |
13 |
400 |
4.0 |
0.033 |
| 09-Dec-2005 |
6 |
200 |
2.0 |
0.030 |
| 10-Dec-2005 |
24 |
610 |
6.1 |
0.039 |
| 11-Dec-2005 |
15 |
490 |
4.9 |
0.031 |
| 12-Dec-2005 |
6 |
290 |
2.9 |
0.021 |
| 13-Dec-2005 |
16 |
660 |
6.6 |
0.024 |
| 14-Dec-2005 |
11 |
410 |
4.1 |
0.027 |
| 15-Dec-2005 |
20 |
250 |
2.5 |
0.080 |
| 16-Dec-2005 |
16 |
430 |
4.3 |
0.037 |
| 17-Dec-2005 |
29 |
420 |
4.2 |
0.069 |
| 18-Dec-2005 |
3 |
220 |
2.2 |
0.014 |
| 19-Dec-2005 |
21 |
610 |
6.1 |
0.034 |
| 20-Dec-2005 |
20 |
420 |
4.2 |
0.048 |
| 21-Dec-2005 |
2 |
180 |
1.8 |
0.011 |
| 22-Dec-2005 |
14 |
290 |
2.9 |
0.048 |
| 23-Dec-2005 |
10 |
190 |
1.9 |
0.053 |
| 24-Dec-2005 |
3 |
100 |
1.0 |
0.030 |
An i chart on Misplaced:
Figure 1. i chart - Misplaced Components (mouse
over data points to see more details)
Notice that this chart does not give us any
out of control points. We would decide that
there are no assignable causes in this process.
An i chart on Rate:
Figure 2. i chart - Rate (mouse over data points
to see more details)
Similarly, this chart shows that we have an
assignable cause in the process related to 2
of 3 points in Zone A. Although there is some
information from the size of the subgroup or
area of opportunity, the control limits are
not calculated based on the subgroup size so
this may be erroneous.
A c chart on Misplaced Components:
Figure 3. c chart - Misplaced Components (mouse
over data points to see more details)
If we could assume that the number of PCBs
was fairly constant, we could get by with this
chart. However, our number of PCBs range from
100 to 660. We are on very thin ice if we make
the assumption of equal area of opportunity.
A u chart on Misplaced Components by PCB:
Figure 4. u chart - Misplaced Components per
PCB (mouse over data points to see more details)
The u chart gives us a couple of alerts for
assignable causes. That is the function of control
charts, to find possible areas of improvement.
In this case, it appears that December 15 and
17 may have had something change and we need
to investigate to see if we can find a cause.
We are now taking advantage of all the information
available to us.
u chart on Misplaced Components per 100 PCB:
Figure 5. u chart - Misplaced Components per
100 PCB (mouse over data points to see more
details)
This chart gives the same information as the
chart in Figure 4. The transformation of data
makes it easier to use. It displays a mean of
3.478 instead of .035. It gives numbers on the
y axis that are easily visualized. Notice also
that this chart will not have a lower control
limit less than 0, while the i charts in Figures
1 and 2 have negative control limits. Counts
cannot go negative, so Attribute Charts (u,
c, p, etc.) take this into account. Such an
assumption cannot be made for measurement data.
The i chart was designed for measurement data.
Mechanical Failures
A similar situation is true for the i and p
charts. Consider the following data on mechanical
failures. Rate is Mech_failure/Parts and Rate1
is Rate * 100.
| Date |
Parts |
Mech_Failure |
Rate |
Rate1 |
| 03-Feb-2004 |
6 |
1 |
0.167 |
16.667 |
| 04-Feb-2004 |
25 |
3 |
0.120 |
12.000 |
| 05-Feb-2004 |
6 |
0 |
0.000 |
0.000 |
| 08-Feb-2004 |
4 |
0 |
0.000 |
0.000 |
| 12-Feb-2004 |
28 |
4 |
0.143 |
14.286 |
| 15-Feb-2004 |
39 |
3 |
0.077 |
7.692 |
| 16-Feb-2004 |
52 |
6 |
0.115 |
11.538 |
| 17-Feb-2004 |
22 |
0 |
0.000 |
0.000 |
| 18-Feb-2004 |
36 |
0 |
0.000 |
0.000 |
| 19-Feb-2004 |
42 |
0 |
0.000 |
0.000 |
| 22-Feb-2004 |
3 |
0 |
0.000 |
0.000 |
| 23-Feb-2004 |
30 |
0 |
0.000 |
0.000 |
| 24-Feb-2004 |
2 |
0 |
0.000 |
0.000 |
| 29-Feb-2004 |
56 |
1 |
0.018 |
1.786 |
| 01-Mar-2004 |
41 |
3 |
0.073 |
7.317 |
| 04-Mar-2004 |
13 |
0 |
0.000 |
0.000 |
| 05-Mar-2004 |
37 |
0 |
0.000 |
0.000 |
| 07-Mar-2004 |
33 |
0 |
0.000 |
0.000 |
| 08-Mar-2004 |
1 |
1 |
1.000 |
100.000 |
| 09-Mar-2004 |
19 |
5 |
0.263 |
26.316 |
| 10-Mar-2004 |
24 |
0 |
0.000 |
0.000 |
First, an i chart:
Figure 6. i chart - Mechanical Failure Rate
(mouse over data points to see more details)
And the p chart:
Figure 7. p chart - Mechanical Failure Percent
(mouse over data points to see more details)
This is a case where the i chart gives us alerts
that may be erroneous because we have no subgroup
information in the chart. However, the p chart
gives us an alert 09-Mar that we should pay
some attention to. We can see by hovering over
the point on 08-Mar that we only have one part
on that day.
We are in the business of improving our processes.
If we were simply working to show that our processes
are in control, the i chart might suffice. If
we wish to find areas for improvement, we might
choose a different chart more in line with the
data.
In conclusion, if you can avoid an i chart,
do so.
Look to a u chart if:
