7.4
The c-chart (fixed sample size)
• Often there is greater concern for the total number of nonconformities (or defects) (C)
than the fraction of nonconforming (defective) units (p).
• An inspection unit is the basic unit inspected for nonconformities or defects.
• The inspection unit could be:
– An individual item (e.g., a car, refridgerator, laptop, etc.).
– Multiple individual items grouped together (e.g., 5 cars, 10 refridgerators, 25 laptops,
etc.).
– A basic unit of continuous size (e.g., 10 miles of railway or roadway, 100 square feet of
fabric, etc.).
• Goal: Develop control charts for the total number of nonconformities per inspection unit.
• The c-chart is based on the total number of nonconformities or defects in an inspection unit.
• A nonconforming unit is an inspection unit not satisfying one or more of the specifications
for that product.
• A nonconformity is defined to be a specific occurrence on a unit of production that does
not meet specifications.
• Therefore, any individual unit contained in the inspection unit can have more than one nonconformity.
• Recall: a defective unit is a nonconforming unit that is unfit for usage.
• Usual assumption: The occurrence of nonconformities in samples of constant size is wellmodeled by the Poisson distribution. This implies:
– The number of opportunities or potential locations for nonconformities may be infinitely
large. This, in turn, implies that the probability of occurrence of a nonconformity at any
location be small and constant.
– The definition of an inspection unit is the same for each sample. That is, each inspection unit always represents an identical interval, area, or volume of opportunity for the
occurrence of nonconformities.
• Assume that each of the inspection units are the same size or contain the same quantity of
production units and the occurrence of nonconformities within an inspection unit follows a
Poisson distribution with parameter c.
• Nonconformity (or defect) data are more informative than fraction nonconforming data because there will usually be several types of nonconformities.
• Therefore, we can simultaneously chart each type of nonconformity on individual c-charts as
long as the Poisson conditions above are met for each type.
• Pareto charts can also accompany the c-charts to summarize defects data.
• In practice, the conditions to assume a Poisson distribution are not satisfied exactly. As long
as the departures are not severe, the Poisson distribution works reasonably well.
103
7.4.1
c is Known
• Let Ci be the number of nonconformities in the ith inspection unit.
• Because Ci ∼ Poisson(c), we know µCi = c and σC2 i = c.
• The control limits for the c-chart are:
UCL = c + 3
Centerline = c
(16)
LCL = c − 3
.
• If LCL < 0, them reset LCL = 0.
• As long as ci remains within control limits for each unit and no systematic pattern is evident,
we conclude the process is in control at level c.
• If ci is outside the control limits or a systematic pattern is evident, we conclude the process
has shifted to a new level and is out of control at level c.
7.4.2
c is Unknown
• The control limits can be established by testing the validity of trial control limits based on m
preliminary samples.
Pm
ci
• The estimate of c used in constructing the trial limits is c = i=1 .
m
• The trial control limits are:
UCL = c + 3
Centerline = c
(17)
LCL = c − 3
• Testing the validity of the trial limits is identical to the method discussed for the p-chart.
• If one or more of the ci are outside of the trial control limits, then assignable causes should
be sought. Next, follow the rules used for p-charts.
• Records on the different types of nonconformities that occur can easily be kept with this type
of control chart to help determine which type of defect is most common.
104
Example for c-chart with known c: Newly painted trucks of the same model are inspected, and the
number of paint defects per truck is recorded. When the paint process is in control there are an average of
7 defects per truck. Therefore, we will use c = 7 as a known total number of defects. The data set contains
a truck ID and the number of defects observed on that truck.
SAS Code for c-chart (c known)
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\CCHART1.PDF’;
ODS LISTING ;
OPTIONS NODATE NONUMBER LS=76 PS=54;
******************************************************;
***
c-chart (c known)
***;
*** The response is the number of paint defects
***;
***
per truck (inspection unit)
***;
******************************************************;
DATA trucks;
INPUT truckid $ defects @@;
LINES;
C1 5
C2 4
C3 4
C4 8
C5 7
C6 12
C7 3
C8 11
E4 8
E9 4
E7 9
E6 13
A3 5
A4 4
A7 9
Q1 15
Q2 8
Q3 9
Q9 10
Q4 8
;
/* Specify Expected Number of Nonconformities using u0= Option */
TITLE ’c Chart for Paint Defects on New Trucks (given c=7)’;
SYMBOL1 WIDTH=3 VALUE=DOT;
PROC SHEWHART DATA=trucks;
CCHART defects*truckid=’1’ / u0 = 7 CSYMBOL = c0
TESTS = 1 to 8 LTESTS = 20
ALLLABEL=(truckid)
ZONELABELS TABLETESTS TABLELEGEND;
LABEL truckid = ’Truck ID’
defects = ’Defects’;
RUN;
Example for c-chart with unknown c: Trucks of the same model are to be painted using a new painting
process. Twenty trucks are painted using the new process and are then inspected. The number of paint
defects per truck is recorded. Twenty trucks are used to establish control limits for the total number of
defects per truck (assuming the process is deemed to be in control). The data set contains a truck ID and
the number of defects observed on that truck.
SAS Code for c-chart (c unknown)
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\CCHART2.PDF’;
ODS LISTING;
OPTIONS NODATE NONUMBER LS=76 PS=54;
******************************************************;
***
c-chart (c unknown)
***;
*** The response is the number of paint defects
***;
***
per truck (inspection unit)
***;
******************************************************;
DATA trucks;
INPUT truckid $ defects @@;
LINES;
B1 12
B2
4
B3
4
B4
3
B5
4
D1
2
D2
3
D3
3
D4
2
D9
4
M2
9
M6 13
L3
5
L4
4
L7
6
Z1 15
Z2
8
Z3
9
Z7
6
Z9
8
;
TITLE1 ’c Chart for Paint Defects in New Trucks (c unknown)’;
SYMBOL1 V=dot W=3;
PROC SHEWHART DATA=trucks;
CCHART defects*truckid=’1’ / TESTS = 1 to 8 LTESTS = 20
ALLLABEL=(truckid) ZONELABELS
TABLETESTS TABLELEGEND;
LABEL truckid = ’Truck ID’
defects = ’Defects’;
RUN;
105
SAS output for c chart (c known)
c Chart for Paint Defects on New Trucks (given c=7)
The SHEWHART Procedure
Test Descriptions
Test 1 One point beyond Zone A (outside control limits)
SAS output for c chart (c unknown)
c Chart for Paint Defects in New Trucks (c unknown)
The SHEWHART Procedure
Test Descriptions
Test 1 One point beyond Zone A (outside control limits)
Test 2 Nine points in a row on one side of center line
Test 6 Four out of five points in a row in Zone B or beyond
106
SAS output for c chart (c known)
c Chart for Paint Defects on New Trucks (given c=7)
The SHEWHART Procedure
c Chart Summary for defects
Subgroup
Sample
Size
truckid
C1
C2
C3
C4
C5
C6
C7
C8
E4
E9
E7
E6
A3
A4
A7
Q1
Q2
Q3
Q9
Q4
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
Test 1
-3 Sigma Limits with n=1 for CountLower
Subgroup
Upper
Limit
Count
Limit
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5.000000
4.000000
4.000000
8.000000
7.000000
12.000000
3.000000
11.000000
8.000000
4.000000
9.000000
13.000000
5.000000
4.000000
9.000000
15.000000
8.000000
9.000000
10.000000
8.000000
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
14.937254
Special
Tests
Signaled
1
Test Descriptions
One point beyond Zone A (outside control limits)
SAS output for c chart (c unknown)
c Chart for Paint Defects in New Trucks (c unknown)
The SHEWHART Procedure
c Chart Summary for defects
truckid
B1
B2
B3
B4
B5
D1
D2
D3
D4
D9
M2
M6
L3
L4
L7
Z1
Z2
Z3
Z7
Z9
Subgroup
Sample
Size
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
Test 1
Test 2
Test 6
-3 Sigma Limits with n=1 for CountLower
Subgroup
Upper
Limit
Count
Limit
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Test
One point beyond
Nine points in a
Four out of five
12.000000
4.000000
4.000000
3.000000
4.000000
2.000000
3.000000
3.000000
2.000000
4.000000
9.000000
13.000000
5.000000
4.000000
6.000000
15.000000
8.000000
9.000000
6.000000
8.000000
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
13.669940
Special
Tests
Signaled
6
2
1
Descriptions
Zone A (outside control limits)
row on one side of center line
points in a row in Zone B or beyond
107
7.5
The u-chart (variable sample size)
• If the interval, area, or volume of opportunity for nonconformities varies from sample to
sample, we may standardize the unit size and use the u-chart.
• The average number of nonconformities per inspection unit is u = c/n where c is the
total number of nonconformities and n is the number of inspection units per sample.
• Note: For c-charts, the inspection unit is the primary sampling unit. For u-charts, the
inspection unit represents the standard unit size.
• Let Ci be the total number of nonconformities and ni be the number of inspection units in
the ith sample. Assume Ci ∼ Poisson(ni u).
• Then Ui = Ci /ni is the average number of nonconformities per inspection unit in the ith
sample. Hence,
E(Ui ) =
var(Ui ) =
• Note: ni is not necessarily an integer. For example, if the inspection unit is a square foot,
then n can be 1.5 square feet.
7.5.1
u is Known
• The control Limits for the u chart are
UCL = u + 3
Centerline = u
(18)
LCL = u − 3
• As long as ui remains within control limits for each unit and no systematic pattern is evident,
we conclude the process is in control at level u.
• If ui is outside the control limits or a systematic pattern is evident, we conclude the process
has shifted to a new level and is out of control at level u.
7.5.2
u is Unknown
• When u is not known, it is estimated from observed data.
1. Select m preliminary samples.
2. Our estimate of u is the average of nonconformities per inspection unit:
Pm
Ci
Total number of nonconformities
u = Pi=1
=
m
Total number of inspection units
i=1 ni
108
3. The trial control Limits for the u chart are
UCL = u + 3
Centerline = u
(19)
LCL = u − 3
• If one or more of the ui are outside of the trial control limits, then assignable causes should
be sought. Next, follow the rules used for p-charts.
Example for u-chart with known u: A textile company uses a u-chart to monitor the number
of defects per square meter of fabric. The fabric is spooled onto rolls as it is inspected for defects.
Each roll of fabric is one meter wide and 30 meters in length, and an inspection unit is defined
as one square meter. Thus, there are 30 inspection units per roll of fabric. Three operators were
used during data collection. When the process is in control there is an average of 0.20 defects per
square meter. Therefore, we will use u = 0.20 as a known average number of defects. The data set
contains a roll number, the number of square meters of fabric inspected from that roll (inspection
unit), and the total number of defects observed for that inspection unit.
SAS Code for u-chart (u known)
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\UCHART1.PDF’;
ODS LISTING;
OPTIONS NODATE NONUMBER LS=76 PS=54;
****************************************************;
***
u-chart (u known)
***;
*** The response is the number of fabric defects ***;
***
per square meter
***;
****************************************************;
DATA fabrics1;
INPUT operator roll defects sqmeters @@;
LINES;
1 1 7 30.0 2 2 11 27.6 3 3 15 30.4
1 4 6 34.8 2 5 11 26.0 3 6 15 28.6
1 7 5 28.0 2 8 10 30.2 3 9 8 28.2
1 10 3 31.4 2 11 3 30.3 3 12 14 27.8
1 13 3 27.0 2 14 9 30.0 3 15 7 32.1
1 16 6 34.8 2 17 7 26.5 2 18 5 30.0
3 19 14 31.3 1 20 13 31.6 2 21 11 29.4
3 22 6 28.6 1 23 6 27.5 2 24 9 32.6
3 25 11 31.7
;
TITLE ’u Chart for Fabric Defects per Square Meter (given u=.20)’;
SYMBOL1 V=dot W=3;
PROC SHEWHART DATA=fabrics1 ;
UCHART defects*roll=’1’ / SUBGROUPN = sqmeters ALLLABEL=(operator)
U0 = 0.20 USYMBOL = u0 TABLE ;
LABEL roll = ’Roll of Fabric’;
RUN;
109
Example for u-chart with unknown u: The description is the same as above except that u is
unknown. Various size inspection units from twenty-five rolls are used to establish control limits
for the total number of defects per square meter of fabric (assuming the process is deemed to be in
control).
SAS Code for u-chart (u unknown)
DM ’LOG; CLEAR; OUT; CLEAR;’;
ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\UCHART2.PDF’;
OPTIONS NODATE NONUMBER;
****************************************************;
***
u-chart (u unknown)
***;
*** The response is the number of fabric defects ***;
***
per square meter
***;
****************************************************;
DATA fabrics2;
INPUT operator roll defects sqmeters @@;
LINES;
1 1 7 30.0 2 2 11 27.6 3 3 15 30.4
1 4 6 34.8 2 5 11 26.0 3 6 15 28.6
1 7 5 28.0 2 8 10 30.2 3 9 8 28.2
1 10 3 31.4 2 11 3 30.3 3 12 14 27.8
1 13 3 27.0 2 14 9 30.0 3 15 7 32.1
1 16 6 34.8 2 17 7 26.5 2 18 5 30.0
3 19 14 31.3 1 20 13 31.6 2 21 11 29.4
3 22 6 28.6 1 23 6 27.5 2 24 9 32.6
3 25 11 31.7
;
TITLE ’u Chart for Fabric Defects per Square Meter (u unknown)’;
SYMBOL1 V=dot W=3;
PROC SHEWHART DATA=fabrics2 ;
UCHART defects*roll=’1’ / SUBGROUPN = sqmeters
ALLLABEL=(operator) TABLE ;
LABEL roll = ’Roll of Fabric’;
RUN;
110
SAS output for u-chart (u known)
u Chart for Fabric Defects per Square Meter (given u=.20)
The SHEWHART Procedure
SAS output for u-chart (u unknown)
u Chart for Fabric Defects per Square Meter (u unknown)
The SHEWHART Procedure
111
SAS output for c chart (c known)
u Chart for Fabric Defects per Square Meter (given u=.20)
The SHEWHART Procedure
u Chart Summary for defects
roll
Subgroup
Sample
Size
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
30.0000
27.6000
30.4000
34.8000
26.0000
28.6000
28.0000
30.2000
28.2000
31.4000
30.3000
27.8000
27.0000
30.0000
32.1000
34.8000
26.5000
30.0000
31.3000
31.6000
29.4000
28.6000
27.5000
32.6000
31.7000
-3 Sigma Limits for Count per UnitSubgroup
Lower
Count
Upper
Limit
per Unit
Limit
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.23333333
0.39855072
0.49342105
0.17241379
0.42307692
0.52447552
0.17857143
0.33112583
0.28368794
0.09554140
0.09900990
0.50359712
0.11111111
0.30000000
0.21806854
0.17241379
0.26415094
0.16666667
0.44728435
0.41139241
0.37414966
0.20979021
0.21818182
0.27607362
0.34700315
0.44494897
0.45537696
0.44333213
0.42742941
0.46311741
0.45087260
0.45354628
0.44413654
0.45264558
0.43942607
0.44373334
0.45445668
0.45819889
0.44494897
0.43680111
0.42742941
0.46062335
0.44494897
0.43980823
0.43866719
0.44743583
0.45087260
0.45584086
0.43497813
0.43829044
SAS output for c chart (c unknown)
u Chart for Fabric Defects per Square Meter (u unknown)
The SHEWHART Procedure
u Chart Summary for defects
roll
Subgroup
Sample
Size
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
30.0000
27.6000
30.4000
34.8000
26.0000
28.6000
28.0000
30.2000
28.2000
31.4000
30.3000
27.8000
27.0000
30.0000
32.1000
34.8000
26.5000
30.0000
31.3000
31.6000
29.4000
28.6000
27.5000
32.6000
31.7000
---3 Sigma Limits for Count per Unit-Subgroup
Lower
Count
Upper
Limit
per Unit
Limit
0.00000000
0.00000000
0.00000000
0.01511065
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00071343
0.00000000
0.00000000
0.00000000
0.00000000
0.00386365
0.01511065
0.00000000
0.00000000
0.00025480
0.00162417
0.00000000
0.00000000
0.00000000
0.00605141
0.00207630
0.23333333
0.39855072
0.49342105
0.17241379
0.42307692
0.52447552
0.17857143
0.33112583
0.28368794
0.09554140
0.09900990
0.50359712
0.11111111
0.30000000
0.21806854
0.17241379
0.26415094
0.16666667
0.44728435
0.41139241
0.37414966
0.20979021
0.21818182
0.27607362
0.34700315
112
0.58201323
0.59452788
0.58007285
0.56098796
0.60381722
0.58912219
0.59233088
0.58103822
0.59124994
0.57538517
0.58055434
0.59342345
0.59791449
0.58201323
0.57223496
0.56098796
0.60082409
0.58201323
0.57584381
0.57447444
0.58499771
0.58912219
0.59508461
0.57004720
0.57402231
7.5.3
Operating Characteristic Function
• The Operating Characteristic Curve (OCC) for both the c-chart and the u-chart are based
on the Poisson distribution.
• For the c-chart, the OCC is a plot of β against c, the true mean total number of defects per
inspection unit.
βc = P (LCL < Ci < UCL|c)
= P (Ci < UCL|c) − P (Ci ≤ LCL|c)
= P (Ci < bUCLc|c) − P (Ci ≤ bLCLc|c)
bUCLc
=
X e−c ci
i!
i=dLCLe
where Ci ∼ Poisson(c), and bUCLc is the largest integer ≤ UCL and dUCLe is the smallest
integer ≥ LCL.
• For the u-chart, the OCC is a plot of β against u, the true mean number of defects per
inspection unit for sample size n. Because Ui = Ci /n:
βu = P (LCL < Ui < UCL|u)
= P (Ci < nUCL|u) − P (Ci ≤ nLCL|u)
= P (Ci < bnUCLc|c) − P (Ci ≤ bnLCLc|c)
bnUCLc
=
X
i=dnLCLe
e−nu (nu)i
i!
113