Control Charts for Attributes
By the attribute method we mean the measurement of quality through noting the
presence or absence of some characteristic in each of the units, and counting how
many units do not posses the quality characteristic. The advantage of the attribute
method is that a single chart can be set up for several characteristics, whereas a
variables chart must be set up for each of the characteristics with an accompanying
chart for controlling variability.
p-Chart for Fraction Nonconforming
This is also known as the proportion chart. The p-chart configuration is intended to
evaluate the process in terms of the proportion or fraction of the total units in a sample
in which a designated classification event occurs. This designated classification event
may be a deviation more than the specified on a measurement scale, quasimeasurement scale, go or not-go gauge, judgment, etc. It could also be a
nonconformity, defect, blemish, presence or absence of some characteristic, etc. The
classification may also be based on several characteristics. Instead of proportions, if
percents are used, then the p-chart will stand for percent chart.
Let p stands for the fraction nonconforming of the process and be the sample
fraction nonconforming computed as the ratio of the number of nonconforming units
d to the sample size n. That is,
parameters n and p ie
= d/n. Let d follows a binomial distribution with
It is further known that the mean and variance of are p and p(1-p)/n respectively. If
the true value of p is known, the control limits become
with the central line being at p. Here p could be a standard value p'.
Suppose that the true fraction nonconforming is unknown. As usual, it is assumed that
the total number of units tested from the process is subdivided into m rational
subgroups consisting of n1, n2, n3...ni...nm units respectively and a value of the
proportion defective is computed for each subgroup. For convenience, one assumes
that the subgroup sizes are all equal. If di is the number of defectives found in the ith
subgroup, then the estimate of p is
is
i
= di/n. The average of various
. The control limits are set at
i
values
For example, consider the following data on the number of defectives obtained for 50
subgroups of 100 resistors drawn from a process.
Table 3.12 Nonconforming resistors in various subgroups
i
di
1
2
3
4
5
6
7
8
9
10
0
0
2
0
1
0
2
2
1
1
i
0.00
0.00
0.02
0.00
0.01
0.00
0.02
0.02
0.01
0.01
i
di
11
12
13
14
15
16
17
18
19
20
0
2
1
1
1
0
0
0
2
3
i
0.00
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.02
0.03
The table given below also gives the
limits are found as
i
di
21
22
23
24
25
26
27
28
29
30
1
2
0
1
1
0
0
1
0
0
i
i
0.01
0.02
0.00
0.01
0.01
0.00
0.00
0.01
0.00
0.00
i
di
31
32
33
34
35
36
37
38
39
40
1
3
0
1
2
2
0
2
2
1
i
0.01
0.03
0.00
0.01
0.02
0.02
0.00
0.02
0.02
0.01
values. The value of
i
di
41
42
43
44
45
46
47
48
49
50
1
1
3
2
1
1
0
0
0
2
i
0.01
0.01
0.03
0.02
0.01
0.01
0.00
0.00
0.00
0.02
is 0.01. The control
.
Figure 3.44 provides the MINITAB p-chart output for the above data.
Figure 3.44 MINITAB p-chart Output
If the computed value for LCL is negative, it is set at zero. This means that there is no
'control' exercised to detect any quality improvement. If the subgroup sizes are
unequal, then p is estimated as
and the (varying) control limits are given by
Alternatively, an 'average' sample size
could be used. One can also
plot the standardised value of pi against the control limits ±3.
Choice of Subgroup Size
Let p1 be the process average fraction defective (considered as acceptable) and
suppose that we wish to detect a k level shift in a p-chart. That is, the process
fraction defective shifts to an unacceptable level p2 = p1 + k. Let us further suppose
that we wish to fix the probability of detecting the shift at 0.5. Under the normal
approximation to binomial, the upper control limit of the
p-chart based on p = p1 will coincide with p2. Hence one has
From this relation, one obtains a formula for n as:
For example, if the shift of the process to an unacceptable level of 0.02 from the
acceptable level of 0.005 must be detected with 50% probability, the subgroup size
should be about 200. If small subgroup sizes are used and p is small, the observance
of one defective may mean an out-of-control signal. In order to avoid this, it may be
desirable to have a large sample size for the p-chart.
Sometimes it may be desirable to have a lower control limit greater than zero in order
to look for samples that contain no defectives or to detect quality improvement. That
is, it is desired that
If p is small, obviously, the value of n should be very large. For example, for p = 0.01,
the minimum subgroup size becomes 891.
Operating Characteristic Curve
1. With only an upper control limit:
The OC function giving the probability that a value of p=pi (or di) falls below the UCL
[or int(nUCL)] is given by
where n is the subgroup size.
2. With both lower and upper control limits:
The OC function giving the probability that pi (or di) falls within the LCL and UCL is
given by
For very large subgroup sizes, LCL is also used as an action limit since a very low
number of defectives may imply the inaccurate use of gages etc. The LCL is
necessary to detect a quality improvement. If only quality degradation need be
detected, the OC function based on the upper control limit should suffice.
MINITAB macros can be used to draw the OC curves of a p-chart. For example, the
OC curves of the p-chart having n=50, LCL=1 and UCL=18 are given in Figures 3.45
and 3.46.
Figure 3.45 OC Curve With Both UCL and LCL
Figure 3.46 OC Curve With Only UCL
np-chart
The np-chart is intended to evaluate the process in terms of the total number of units
in a sample in which a given classification event occurs.
The np-chart is essentially a p-chart, the only difference being the observed number of
defectives is directly plotted instead of the observed proportion defective. If p is the
proportion defective, then d, the number of defectives in the subgroup size n, follows
a binomial distribution whose expected value is np with standard
deviation
. Here p could be a standard value. When no standards are
available, one uses the
estimate and draws the control limits
at
. The central line is drawn at n . The OC function of
the np chart is similar to that of the p-chart and on the x-axis one plots the d values
instead of p values.
c-chart for Counts
By the term area of opportunity, we mean a unit or a portion of material, process,
product or service in which one or more designated events occur. The term is
synonymous with the term unit and is usually preferred where there is no natural unit,
eg continuous length of cloth.
By defect, we mean the departure of a characteristic from its prescribed specification
level that will render the product or service unfit to meet the normal usage
requirements. By nonconformity, we mean a product which may not meet the
specification requirement but may meet the usage requirements. For example, dirt in a
block of cheese is a defect but underweight is a nonconformity.
By c or count, we mean the number of events of a given classification occurring in a
sample. More than one such event may occur in a desired area of opportunity.
The c-chart or count chart is a configuration designed to evaluate the process in terms
of the count of events (eg. defect or nonconformity) of a given classification occurring
in a sample.
We assume that the number of nonconformities d follows a Poisson distribution
whose mean and variance are equal to the parameter c. That is, d follows:
.
The control chart for count d with three sigma limits is therefore
c  3c,
the central line being c. If LCL is less than zero, it is set at zero. Here c could be a
standard value. In its absence, c is estimated as the average number of
nonconformities in a sample, say , and the control limits are set at  3 .
Consider the following table showing the number of nonconformities observed in a
sample of 20 subgroups of cellular phones of five each. Here C1 stands for subgroup
number and C2 stands for the number of events (defects) that occured in the given
area of opportunity (phone).
Table 3.13 Number of Nonconformities in various subgroups
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
The value of
C1 C2 Row
1 3 21
1 0 22
1 1 23
1 1 24
1 0 25
2 1 26
2 0 27
2 0 28
2 1 29
2 0 30
3 0 31
3 2 32
3 0 33
3 1 34
3 2 35
4 2 36
4 0 37
4 0 38
4 0 39
4 2 40
C1 C2 Row
5 0 41
5 1 42
5 2 43
5 2 44
5 1 45
6 0 46
6 0 47
6 2 48
6 1 49
6 2 50
7 0 51
7 0 52
7 0 53
7 0 54
7 0 55
8 1 56
8 0 57
8 1 58
8 0 59
8 0 60
C1 C2 Row
9
1 61
9
1 62
9
1 63
9
2 64
9
2 65
10 0 66
10 1 67
10 0 68
10 1 69
10 0 70
11 3 71
11 0 72
11 2 73
11 0 74
11 1 75
12 1 76
12 0 77
12 0 78
12 0 79
12 0 80
C1 C2 Row
13 1 81
13 2 82
13 1 83
13 2 84
13 1 85
14 0 86
14 0 87
14 2 88
14 1 89
14 0 90
15 1 91
15 2 92
15 1 93
15 1 94
15 1 95
16 0 96
16 1 97
16 0 98
16 0 99
16 1 100
C1
17
17
17
17
17
18
18
18
18
18
19
19
19
19
19
20
20
20
20
20
C2
1
1
2
0
2
2
0
1
2
2
0
0
2
0
4
1
0
0
1
0
is 84/20 = 4.2. The control limits are then found as
4.2  3 or
0 to 10.4
and the central line is set at 4.2. One plots the total number of defects found in each
subgroup thereafter in the c-chart. While using a c-chart, a signal for a special cause
may require further analysis using a cause and effect diagram.
Figure 3.48 is the MINITAB output of the c-chart for the above data. It should be
noted that MINITAB has no provision to input subgroup codes for drawing a c-chart.
Figure 3.48 MINITAB c-chart Output
OC Function
1. With UCL only:
If action is taken only with the upper control limit (ie d exceeding UCL), the OC
function giving the probability of d being less than UCL is
since d follows the Poisson distribution with parameter c. To obtain the OC curve, one
computes Pa as above for a given value of c, and a plot of (c, Pa) will yield the OC
curve. It can be noted that the above OC function is similar to the OC function of a
single sampling plan (discussed in Chapter 4) with acceptance number Ac =
int(UCL). The sample size of the single sampling plan cannot be interpreted directly.
2. With both LCL and UCL:
The OC function of the c-chart with both LCL and UCL giving the probability of d
falling within LCL and UCL is
.
It is easy to evaluate the OC function of the c-chart as
While using calculators to find Pa, the recurrence relation or normal approximation
may be useful.
Figures 3.49 and 3.50 show the OC curves of a c-chart obtained using MINITAB
macros.
Figure 3.49 OC Curve With Only UCL
Figure 3.50 OC Curve With Both UCL and LCL
u-chart
The u-chart or count per unit chart is a configuration to evaluate the process in terms
of average number of events of a given classification per unit area of opportunity. The
u-chart is convenient for a product composed of units whose inspection covers more
than one characteristic such as dimension checked by gages, electrical and mechanical
characteristics checked by tests, and visual defects observed by eye. Under these
conditions, independent defects may occur in one unit of product and a preferred
quality measure is to count all defects observed and divide by the number of units
inspected to give a value for defects per unit, rather than a value for the fraction
defective. Here only the independent defects are to be counted. The u-chart is
particularly useful for products such as textiles, wire, sheet materials, etc, which are
continuous and extensive. Here the opportunity for defects/nonconformities is large
even though the chance of a defect at one spot is small.
The total number of units tested is subdivided into m rational subgroups of size n
each. For each subgroup, a value of u, the defects per unit, is computed. The average
number of defects is found as
Assuming that the number of defects follows the Poisson distribution, the control
limits of the u-chart are given by,
For unequal subgroups,
is found as
where ni is the ith subgroup size and ui is the number of defects per unit in the ith
subgroup. Here n1 , n2 ... need not be whole numbers, eg the length of the cloth
inspected may be 2.4 m. The control limits are set at
One can also use the standardised variable and plot as an I-chart. If a c-chart is drawn
with unequal subgroup sizes, the control limits are set at
For the cellular phone data considered earlier, the u-chart drawn using MINITAB is
given in Figure 3.51 (MINITAB requires the di values to be input directly rather than
the ui values).
Figure 3.51 MINITAB u-chart Output
The OC function of the u-chart is similar to the OC function of the c-chart. One uses
the relation c = nu and the results will follow.
Demerit Charts and Defect Classification
The c- and u-charts give equal importance to various types of defects irrespective of
their degree of seriousness. Defects are usually classified into the following four types
and charts (c or u) are separately drawn for each type of defect.
Class I - Very Serious - leading directly to severe injury or catastrophic economic
loss.
Class II - Serious - leading to significant injury or significant economic loss.
Class III - Major - leading to major problems in the normal usage.
Class IV- Minor - leading to minor problems in the normal usage.
Eg in textile garments, easy inflammability is a Class I defect: shrinkage or colour
fade is a Class II defect, weaker weft yarn is Class III, and dirt is Class IV.
By demerit, we mean a weighting given to a defect type to obtain a weighted quality
score. Let there be k defect types. Also let dj be the number of defects due to the jth
type and wj is the demerit score for the jth defect type. Then the statistic
can be computed for each subgroup. Then
will form the
central line where
is the mean of the D's computed for various subgroups and
is the mean number of defects due to the jth type. The control limits become
.
In some applications, the demerit score will be distinguished from a quality score. By
a quality score, we mean a numerical indicator to measure the relative quality of the
incoming material, operations, in-process or final product or service etc. This score is
often useful for the relative evaluation of quality. For example, if a large company is
manufacturing a number of products for both local and export markets in different
plants, a relative quality evaluation will be done with the quality scores. If qj is the
quality score for each defect-product type (say), the control limits will be set at
with the central line being drawn at
management, this chart will be useful.
For quality evaluation by top