Chapter 15
Statistical
Quality Control
(Revised 8/10/04)
To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved.
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Statistical Process
Control
• Take periodic samples from process
• Plot sample points on control chart
• Determine if process
UCL
is within limits
• Prevent quality
problems
LCL
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Variation
 Common Causes
Variation inherent in a process
Can be eliminated only through
improvements in the system
 Special Causes
Variation due to identifiable factors
Can be modified through operator or
management action
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Types of Data
 Attribute data
 Product characteristic
evaluated with a discrete choice
 Good/bad, yes/no
 Variable data
 Product characteristic that
can be measured
 Length, size, weight, height,
time, velocity
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SPC Applied to
Services
 Nature of defect is different in
services
 Service defect is a failure to meet
customer requirements
 Monitor times, customer
satisfaction
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Service Quality Examples
 Hospitals
Timeliness, responsiveness,
accuracy of lab tests
 Grocery Stores
Check-out time, stocking, cleanliness
 Airlines
Luggage handling, waiting times,
courtesy
 Fast food restaurants
Waiting times, food quality,
cleanliness, employee courtesy
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Service Quality Examples
 Catalog-order companies
Order accuracy, operator
knowledge and courtesy,
packaging, delivery time,
phone order waiting time
 Insurance companies
Billing accuracy, timeliness of claims
processing, agent availability and
response time
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Control Charts
 Graph establishing process control
limits
 Charts for variables
Mean (x-bar), Range (R)
 Chart for attributes
P Chart
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Process Control Chart
Out of control
Upper
control
limit
Process
average
Lower
control
limit
1
Figure 15.1
2
3
4
5
6
7
8
9
10
Sample number
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A Process is In Control if
1. No sample points outside limits
2. Most points near process average
3. About equal number of points
above & below centerline
4. Points appear randomly
distributed
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Development of
Control Chart
 Based on in-control data
 If non-random causes present,
find the special cause and
discard data
 Correct control chart limits
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Control Chart for
Attributes
 p Charts
Calculate percent defectives in sample
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p-Chart
UCL = p + zp
LCL = p - zp
where
z = the number of standard deviations from
the process average
p = the sample proportion defective; an
estimate of the process average
p = the standard deviation of the sample
proportion
p =
p(1 - p)
n
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The Normal Distribution
95%
99.74%
-3
-2
-1
=0
1
2
3
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Control Chart Z Values
 Smaller Z values make more
sensitive charts
 Z = 3.00 is standard
 Compromise between sensitivity
and errors
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
NUMBER OF
DEFECTIVES
PROPORTION
DEFECTIVE
6
0
4
:
:
18
200
.06
.00
.04
:
:
.18
Example 15.1
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
NUMBER OF
DEFECTIVES
6
0
4
:
:
18
200
PROPORTION
DEFECTIVE
.06
.00
total defectives
p = .04
total sample observations
:
= 200: / 20(100)
= 0.10
.18
Example 15.1
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
NUMBER OF
DEFECTIVES
PROPORTION
DEFECTIVE
p = 0.10
6
.06
0
0.10(1 - 0.10)
p(1.00
- p)
UCL = p + z
= 0.10 + 3
100
n
4
.04
:
UCL := 0.190
:
0.10(1 - 0.10)
p(1 - p):
LCL
= 0.10 - 3
18= p - z
100
n.18
200= 0.010
LCL
Example 15.1
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p-Chart
0.20
UCL = 0.190
0.18
0.16
Proportion defective
0.14
0.12
0.10
p = 0.10
0.08
0.06
0.04
0.02
LCL = 0.010
2
4
6
8
10
12
Sample number
14
16
18
20
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Control Charts for
Variables
 Mean chart ( x -Chart )
Uses average of a sample
 Range chart ( R-Chart )
Uses amount of dispersion in
a sample
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Range ( R- ) Chart
UCL = D4R
LCL = D3R
R
R= k
where
R = range of each sample
k = number of samples
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SAMPLE SIZE
n
FACTOR FOR x-CHART
A2
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.44
0.11
0.99
0.77
0.55
0.44
0.22
0.11
0.00
0.99
0.99
0.88
FACTORS FOR R-CHART
D3
D4
Range ( R- ) Chart
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
0.36
0.38
0.39
0.40
0.41
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
1.64
1.62
1.61
1.61
1.59
Table 15.1
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R-Chart Example
OBSERVATIONS (SLIP-RING DIAMETER, CM)
SAMPLE k
1
2
3
4
5
x
R
1
2
3
4
5
6
7
8
9
10
5.02
5.01
4.99
5.03
4.95
4.97
5.05
5.09
5.14
5.01
5.01
5.03
5.00
4.91
4.92
5.06
5.01
5.10
5.10
4.98
4.94
5.07
4.93
5.01
5.03
5.06
5.10
5.00
4.99
5.08
4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99
4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
50.09
1.15
Example 15.3
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R-Chart Example
UCL = D4R = 2.11(0.115) = 0.243
R
1.15
R=
=
= 0.115
OBSERVATIONS
(SLIP-RING
DIAMETER, CM)
k
10
LCL = D
3R = 0(0.115) = 0
SAMPLE0.28
k –
1
2
3
4
5
x
R
Range
1
2
3
4
5
6
7
8
9
10
0.24 –
0.20 –
0.16 –
0.12 –
0.08 –
0.04 –
0–
5.02 5.01 4.94
5.01
UCL 5.03
= 0.2435.07
4.99 5.00 4.93
5.03 4.91 5.01
0.115 5.03
4.95R =4.92
4.97 5.06 5.06
5.05 5.01 5.10
5.09 5.10 5.00
5.14 5.10 4.99
LCL = 0
5.01
|
| 4.98| 5.08
|
1
2
3
4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07
|
|
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99|
4
5
6
7
Sample number
4.98 0.08
5.00 0.12
4.97 0.08
4.96 0.14
4.99 0.13
5.01 0.10
5.02 0.14
5.05 0.11
5.08 0.15
5.03
|
| 0.10|
50.09
8
91.1510
Example 15.3
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x-Chart Calculations
x1 + x2 + ... xk
=
x=
k
=
UCL = x + A2R
=
LCL = x - A2R
where
=
x = the average of the sample means
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x-Chart Example
OBSERVATIONS (SLIP-RING DIAMETER, CM)
SAMPLE k
1
2
3
4
50.095.01 4.94 4.99
=1 x
5.02
x=
=
= 5.01 cm
k
2
5.01
10 5.03 5.07 4.95
5
x
R
4.96 4.98
4.96 5.00
3
4.99 5.00 4.93 4.92 4.99 4.97
4
5.03 4.91 5.01 4.98 4.89 4.96
=
UCL5 = x + A2R4.95
= 5.01
= 5.08
4.92+ (0.58)(0.115)
5.03 5.05 5.01
4.99
6
4.97 5.06 5.06 4.96 5.03 5.01
5.01- (0.58)(0.115)
5.10 4.96 4.99
5.02
LCL7 = x= - A2R5.05
= 5.01
= 4.94
8
5.09 5.10 5.00 4.99 5.08 5.05
9
5.14 5.10 4.99 5.08 5.09 5.08
10
5.01 4.98 5.08 5.07 4.99 5.03
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
50.09
1.15
Example 15.4
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x-Chart Example
5.10 –
5.08 –
5.06 –
SAMPLE k
UCL = 5.08
OBSERVATIONS (SLIP-RING DIAMETER, CM)
1
2
3
4
5.04 – 50.09
=1 x
4.94 4.99
x=
= 5.02 5.01= 5.01
cm
k5.02 – 5.0110=5.03 5.07 4.95
2
5
x
Mean
4.96 4.98
4.96 5.00
= 5.01
3
4.99 x5.00
4.93 4.92 4.99 4.97
5.00 – 5.03 4.91
4
5.01 4.98 4.89 4.96
=
UCL5 = x +
A R = 5.01 + (0.58)(0.115)
= 5.08
5.03 5.05 5.01
4.99
4.98 –2 4.95 4.92
6
4.97 5.06 5.06 4.96 5.03 5.01
5.01- (0.58)(0.115)
5.10 4.96 4.99
5.02
LCL7 = x= -4.96
A2–R5.05
= 5.01
= 4.94
8
5.09 LCL
5.10
5.00 4.99 5.08 5.05
= 4.94
4.94 –
9
5.14 5.10 4.99 5.08 5.09 5.08
10
5.08 5.07 4.99 5.03
4.92 – 5.01 4.98
|
1
Example 15.4
|
2
|
3
|
|
|
|
4
5
6
7
Sample number
R
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
| 50.09
|
|1.15
8
9
10
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Using x- and R-Charts
Together
 Each measures the process
differently
 Both process average and variability
must be in control
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Sample Size Determination
 Attribute control charts
50 to 100 parts in a sample
 Variable control charts
2 to 10 parts in a sample
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Process Capability
Process limits (The “Voice of the Process” or
The “Voice of the Data”) - based on natural
(common cause) variation
•
Tolerance limits (The “Voice of the Customer”)
– customer requirements
•
Process Capability – A measure of how
“capable” the process is to meet customer
requirements; compares process limits to
tolerance limits
•
Process Capability
 Range of natural variability in process
 Measured with control charts.
 Process cannot meet specifications if
natural variability exceeds tolerances
 3-sigma quality
 Specifications equal the process control
limits.
 6-sigma quality
 Specifications twice as large as control
limits
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Process Capability
Design
Specifications
(a) Natural variation
exceeds design
specifications; process
is not capable of
meeting specifications
all the time.
Process
Design
Specifications
(b) Design specifications
and natural variation the
same; process is capable
of meeting specifications
most the time.
Process
Figure 15.5
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Process Capability
Design
Specifications
(c) Design specifications
greater than natural
variation; process is
capable of always
conforming to
specifications.
Process
Design
Specifications
(d) Specifications greater
than natural variation,
but process off center;
capable but some output
will not meet upper
specification.
Process
Figure 15.5
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Process Capability
Measures
Process Capability Index
Cpk = minimum
=
x - lower specification limit
,
3
=
upper specification limit - x
3
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Computing Cpk
Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
Cpk = minimum
= minimum
=
x - lower specification limit
,
3
=
upper specification limit - x
3
8.80 - 8.50 9.50 - 8.80
,
3(0.12)
3(0.12)
= 0.83
Example 15.7
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Interpreting the Process
Capability Index
Cpk < 1
Not Capable
Cpk > 1
Capable at 3
Cpk > 1.33
Capable at 4
Cpk > 1.67
Capable at 5
Cpk > 2
Capable at 6