MGT 6421
Quality Management II
Control Charts for Attributes

Our objectives for this section are to learn how to use control charts to
monitor discrete data. We want to learn the assumptions behind the
charts, their application, and their interpretation.

Here we are counting either the number of nonconforming items, or
the number of nonconformities. If we are counting the number of
nonconforming items (i.e., the number of "bad" parts) in a sample, the
charts used are referred to as binomial count charts. If instead we are
counting the number of nonconformities (for example, the number of
flaws on the surface), the charts are referred to as area of opportunity
charts.

Procedure for using Binomial Count Charts:
1. Determine what constitutes a nonconforming part.
2. At predetermined, even intervals, take samples of size n. The
sample size must be large enough so that some nonconforming items
are likely to be included in the sample.
3. If the process requires that samples of different size be taken (i.e.,
n changes from sample to sample), then use a p chart. Otherwise, use
an np chart.
4.
a. When using an np chart, plot the number of nonconforming
items in the sample, denoted x.
b. When using a p chart, plot the proportion of nonconforming
items in the sample, x/n.
Attribute Control Charts - 1
MGT 6421
Quality Management II
5. After collecting a sufficient number of samples, k (k>20) compute
the control limits for the appropriate chart (see the table on page 5 for
the appropriate control limit calculations). The following additional
calculation will be necessary:
k
 xi
p
i 1
k
 ni
i 1
 the total number of defective items in all the samples taken 
=

the total number of items sampled


If you are using a p chart, the control limits will change each time n
changes.
6. If any points fall outside of the control limits, conclude that the
process is out of control, and begin a search for an assignable or
special cause. When the special cause is identified, remove that point
and return to step 5 to re-evaluate the remaining points.
7. If all the points are within limits, conclude that the process is in
control, and use the calculated limits for future monitoring of the
process.
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MGT 6421

Quality Management II
Procedure for using Area of Opportunity Charts:
1. Determine what type of nonconformity to chart.
2. At predetermined, even intervals, count the number of
nonconformities on a sampled item.
3. If the process requires that the area being inspected changes from
sample to sample, then use a u chart. Otherwise, use a c chart.
4.
a. When using a c chart, plot the total number of
nonconformities, denoted c.
b. When using a u chart, plot the number of nonconformities
per unit area, u = c/a.
5. After collecting a sufficient number of samples, k (k>20) compute
the control limits for the appropriate chart (see the table on page 5 for
the appropriate control limit calculations). One of the following
additional calculations will be necessary:
k
k
 ci
 ci
c  i 1
k
(for the c chart)
or
u
i 1
k
(for the u chart).
 ai
i 1
If you are using a u chart, the control limits will change each time ai
changes.
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MGT 6421
Quality Management II
6. If any points fall outside of the control limits, conclude that the
process is out-of-control, and begin a search for an assignable or
special cause. When the special cause is identified, remove that point
and return to step 5 to re-evaluate the remaining points.
7. If all the points are within limits, conclude that the process is in
control, and use the calculated limits for future monitoring of the
process.

How to handle unequal sample sizes or areas of opportunity:
As mentioned above, when using a p chart or a u chart, the limits
change each time the sample size does. There are a few possible
alternatives to this:
a. One is to compute an average sample size or area, and base an
average control limit on this. In this case, if a point on the chart
falls "close" to the limits (on either side), the true value of the
limit should be calculated using the actual sample size.
b. A second is to compute 2 sets of limits. An outer limit is based on
the minimum sample size or area, and an inner limit is based on
the maximum sample size or area. Any point which falls within
the inner limits is in control. Any point which falls outside the
outer limits is out of control. Any point which falls between the
two sets of limits needs to be checked further by calculating the
true control limit based on the actual sample size.
c. A third is to compute a Z-score using the appropriate mean and
standard deviation, and use limits of 3.
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MGT 6421
Quality Management II
Limits for Attributes Control Charts
Measure
Chart
number of
nonconforming
items per sample
np
Limits

UCL  np  3 np (1  p )

LCL  max 0, np  3 np(1  p)


proportion of
nonconforming
items per sample
number of
nonconformities per
area of opportunity
UCL  p  3
p
p (1  p )
n

p (1  p ) 
LCL  max 0, p  3

n


UCL= c  3 c
c

LCL= max 0, c  3 c


number of
nonconformities per
unit area of opportunity
u
UCL= u  3
u
a

u
LCL= max 0, u  3 
a


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MGT 6421

Quality Management II
Examples: Determine the appropriate control chart and compute its
limits.
a. A line produces precision lenses. Three lenses are selected at
random from those made from each batch of molten glass, and the
number of voids in these lenses is counted and recorded. Recent data
have averaged 7.2 seeds per sample of 3 lenses.
b. An electronics producer is monitoring the quality of the circuit
boards received from one of its suppliers. Samples of size 80 are
taken from consecutive lots, and the number of nonconforming circuit
boards is recorded and plotted. Recent data have given an average of
1.8 nonconforming boards per sample of 80.
c. The same producer in b. above is reviewing the quality of a former
supplier. At the time they used the supplier, they were not using
process control charts, but they were sampling for inspection
purposes. They were using 3 different sample sizes: 70, 80 and 90.
The overall average proportion nonconforming was found to be .02.
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MGT 6421
Quality Management II
Sample number
Number per sample
Number of green
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Total
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Proportion green
MGT 6421
Quality Management II
 A large publisher counts the number of keyboard errors that make their
way into finished books. The number of errors and the number of pages
in the last 26 publications are shown below. Construct the appropriate
control chart for the data and determine if there are any indications of
lack of control.
Book
1
2
3
4
5
6
7
8
9
Errors
49
63
57
33
54
37
38
45
65
Pages
202
232
332
429
512
347
401
412
481
Book
10
11
12
13
14
15
16
17
18
Errors
62
40
21
35
48
50
41
45
51
Pages
770
577
734
455
612
432
538
383
302
Attribute Control Charts - 8
Book
19
20
21
22
23
24
25
26
Errors
49
38
70
55
63
33
14
44
Pages
285
591
310
547
469
652
343
401
MGT 6421
Quality Management II
Selecting the Appropriate Control Chart

We have already discussed the steps in implementing a particular
control chart. Important factors are what variables to monitor, where
in the process to monitor, how to collect the data, and what chart to
use. We will concentrate on what chart to use.

The choice of chart depends upon the type of data that we are
monitoring. In some cases, it also depends upon what kind of
performance we want the chart to achieve.

The choice of control chart also depends to some degree on where we
are in our process control efforts. It is often the case that
organizations need to begin with attribute control charts. As the
processes improve, it becomes harder to find defects. At that point,
variables control charts are implemented in order to find ways to
improve further.
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MGT 6421

Quality Management II
The following decision tree1 can be helpful when determining which
control chart should be used for a given situation.
1
Taken from Muñoz and Nielsen (1991). "SPC: What Data Should I Collect? What
Charts Should I Use?" Quality Progress, Vol. 24, No. 1, 50-52.
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Quality Management II
Examples
a.
Boxes of ball point pens are packaged 12 to a box, and 50 boxes are put
in each carton for shipment. Every 50th carton of pens produced is
fully inspected. Each pen is inspected for 6 quality characteristics:
width of written line, color of written line, straightness of the pen, color
of the finish of the pen, smoothness of finish, and general appearance
of the pen. On average each pen has .0003 imperfections.
b.
Every 10 minutes, José takes 6 baskets of strawberries off the produce
line and weighs each of them.
c.
A bank wishes to track the proportion of ATM transactions per day.
They record both the total number of transactions and the number of
ATM transactions each day. The overall average proportion of ATM
transactions is .25. The number of total transactions varies
significantly, but has always been between 280 and 410.
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MGT 6421
Quality Management II
d.
A service administers 50 customer surveys each week, and asks
customers if they were satisfied with their experience or not. On
average 1 customer per week reports being dissatisfied.
e.
Since you want to take quality personal, you decide to monitor your
promptness to meetings. As a first pass, you decide to record if you are
late or not to meetings. The number of meetings that you attend daily
varies. So each day you record the number of meetings attended, and
the number to which you are late. After 30 days you find that you have
attended 98 meetings and you have been late to 20 of them.
f.
This is a follow-up to part e. Now you decide to go beyond recording
whether you are late or not, to recording the number of minutes you are
late. For every meeting attended over the next 30 days you write the
number of minutes late. This time you attend 85 meetings over the 30day period, and the total minutes late are 150. (NOTE: For this one I
haven’t given you all the information you need to calculate the limits.
Write down which control chart(s) should be used, and then say what
other information would be required to compute the limits.)
Attribute Control Charts - 12