Chapter 18
Statistical Quality Control
LEARNING OBJECTIVES
Chapter 18 presents basic concepts in quality control, with a particular emphasis on statistical
quality control techniques, thereby enabling you to:
1.
2.
3.
4.
5.
Understand the concepts of quality, quality control, and total quality management.
Understand the importance of statistical quality control in total quality management.
Learn about process analysis and some process analysis tools, including Pareto charts,
fishbone diagrams, and control charts.
Learn how to construct x charts, R charts, p charts, and c charts.
Understand the theory and application of acceptance sampling.
CHAPTER OUTLINE
18.1
Introduction to Quality Control
What Is Quality Control?
Total Quality Management
Some Important Quality Concepts
Benchmarking
Just-in-Time Inventory Systems
Reengineering
Six Sigma
Team Building
18.2
Process Analysis
Flowcharts
Pareto Analysis
Cause-and-Effect (Fishbone) Diagrams
Control Charts
18.3
Control Charts
Variation
Types of Control Charts
x Chart
R Charts
p Charts
c Charts
Interpreting Control Charts
18.4
Acceptance Sampling
Single Sample Plan
Double-Sample Plan
Multiple-Sample Plan
Determining Error and OC Curves
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KEY TERMS
acceptance sampling
after-process quality control
benchmarking
c chart
cause-and-effect diagram
centerline
consumer's risk
control chart
double-sample plan
fishbone diagram
flowchart
in-process quality control
Ishikawa diagram
just-in-time inventory systems
lower control limit (LCL)
manufacturing quality
multiple-sample plan
operating characteristic (OC) curve
p chart
Pareto analysis
Pareto chart
process
producer’s risk
product quality
quality
quality circle
quality control
R chart
reengineering
single-sample plan
Six Sigma
team building
total quality management (TQM)
transcendent quality
upper control limit (UCL)
user quality
value quality
x chart
STUDY QUESTIONS
1. The collection of strategies, techniques, and actions taken by an organization to assure themselves
that they are producing a quality product is _____________________________________.
2. Measuring product attributes at various intervals throughout the manufacturing process in an effort to
pinpoint problem areas is referred to as _______________________ quality control.
3. Inspecting the attributes of a finished product to determine whether the product is acceptable, is in
need of rework, or is to be rejected and scrapped is _________________________ quality control.
4. An inventory system in which no extra raw materials or parts are stored for production is called a
_____________________ system.
5. When a group of employees are organized as an entity to undertake management tasks and perform
other functions such as organizing, developing, and overseeing projects, it is referred to as
______________________________________.
6. A ______________________________ is a small group of workers, usually from the same
department or work area, and their supervisor, who meet regularly to consider quality issues.
7. The complete redesigning of a company's core business process is called
________________________________. This usually involves innovation and is often a complete
departure from the company's normal way of doing business.
8. A total quality management approach that measures the capability of a process to perform defect-free
work is called ____________________.
Chapter 18: Statistical Quality Control
333
9. A methodology in which a company attempts to develop and establish total quality management from
product to process by examining and emulating the best practices and techniques used in their
industry is called ____________________________.
10. A graphical method for evaluating whether a process is or is not in a state of statistical control is
called a ____________________________________.
11. A diagram that is shaped like a fish and displays potential causes of one problem is called a
_______________________ or ______________________ diagram.
12. A bar chart that displays a quantitative tallying of the numbers and types of defects that occur with a
product is called a ___________________________________.
13. Two types of control charts for measurements are the ____________ chart and the _____________
chart. Two types of control charts for attribute compliance are the ____________ chart and the
_____________ chart.
14. An x bar chart is constructed by graphing the ____________ of a given measurement computed for a
series of small samples on a product over a period of time.
15. An R chart plots the sample __________________. The centerline of an R chart is equal to the value
of _________.
16. A p chart graphs the proportion of sample items in ________________________ for multiple
samples. The centerline of a p chart is equal to ____________.
17. A c chart displays the number of _______________________ per item or unit.
18. Normally, an x bar chart is constructed from 20 to 30 samples. However, assume that an x bar chart
can be constructed using the four samples of five items shown below:
Sample 1
23
22
21
23
22
Sample 2
21
18
22
19
19
Sample 3
19
20
20
21
20
Sample 4
22
24
18
16
17
The value of A2 for this control chart is _______________.
The centerline value is ___________________.
The value of R is _________________.
The value of UCL is ____________________.
The value of LCL is ________________.
The following samples have means that fall outside the outer control limits
_____________________________. In constructing an R chart from these data, the value of the
centerline is __________________. The value of D3 is ________________ and the value of D4 is
_________________. The UCL of the R chart is ____________________ and the value of LCL is
____________________.
The following samples have ranges that fall outside the outer control limits
_______________________________________.
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19. p charts should be constructed from data gathered from 20 to 30 samples. Suppose, however, that a p
chart could be constructed from the data shown below:
Sample
1
2
3
4
5
6
n
70
70
70
70
70
70
Number out of Compliance
3
5
0
4
3
6
The value of the centerline is ______________________________.
The UCL for this p chart is _____________________________.
The LCL for this p chart is _____________________________.
The samples with sample proportions falling outside the outer control limits are
_______________________________.
20. c charts should be constructed using at least 25 items or units. Suppose, however, that a c chart could
be constructed from the data shown below:
Item
Number
1
2
3
4
5
6
7
Number of
Nonconformities
3
2
2
4
0
3
1
The value of the centerline for this c chart is ____________.
The value of UCL is _________________ and the value of LCL is _________________.
21. A process is considered to be out of control if ___________ or more consecutive points occur on one
side of the centerline of the control chart.
22. Four possible causes of control chart abnormalities are (at least eight are mentioned in the text)
_____________, _______________, _______________, and _______________.
23. Suppose a single sample acceptance sampling plan has a c value of 1, a sample size of 10, a p0 of .03,
and a p1 of .12. If the supplier really is producing 3% defects, the probability of accepting the lot is
______________ and the probability of rejecting the lot is ______________. Suppose, on the other
hand, the supplier is producing 12% defects. The probability of accepting the lot is
____________________ and the probability of rejecting the lot is ___________________.
24. The Type II error in acceptance sampling is sometimes referred to as the
__________________________ risk. The Type I error in acceptance sampling is sometimes referred
to as the _____________________________ risk.
25. Using the data from question 22, the producer's risk is ___________________________ and the
consumer's risk is ________________________. Assume that 3% defects is acceptable and 12%
defects is not acceptable.
Chapter 18: Statistical Quality Control
335
26. Suppose a two-stage acceptance sampling plan is undertaken with c1 = 2, r1 = 6, and c2 = 7. A sample
is taken resulting in 4 rejects. A second sample is taken resulting in 2 rejects. The final decision is to
__________________ the lot.
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ANSWERS TO STUDY QUESTIONS
1.
Quality Control
2.
In-Process
16. Noncompliance, p (average
proportion)
17. Nonconformances
3.
After-Process
4. Just-in-Time
18. 0.577, 20.35, 4.0, 22.658, 18.042,
None, 4.0, 0, 2.115, 8.46, 0.00, None
5. Team Building
19. .05, .128, .000, None
6.
Quality Circle
20. 2.143, 6.535, 0.00
7.
Reengineering
21. 8
8.
Six Sigma
9.
Benchmarking
10.
Control Chart
22. Changes in the Physical Environment,
Worker Fatigue, Worn Tools, Changes
in Operators or Machines,
Maintenance, Changes in Worker
Skills, Changes in Materials, Process
Modification
11.
Fishbone, Ishikawa
23. .9655, .0345, .6583, .3417
12.
Pareto Chart
24. Consumer’s, Producer’s
13.
x , R, p, c
14.
Means
25. .0345, .6583
26. Accept
15.
Ranges, R
SOLUTIONS TO ODD-NUMBERED PROBLEMS IN CHAPTER 18
18.5
x1 = 4.55, x 2 = 4.10, x 3 = 4.80, x 4 = 4.70,
x 5 = 4.30, x 6 = 4.73, x 7 = 4.38
R1 = 1.3, R2 = 1.0, R3 = 1.3, R4 = 0.2, R5 = 1.1, R6 = 0.8, R7 = 0.6
x = 4.51
R = 0.90
For x Chart:
Since n = 4, A2 = 0.729
Centerline:
x = 4.51
UCL:
x + A2 R = 4.51 + (0.729)(0.90) = 5.17
LCL:
x – A2 R = 4.51 – (0.729)(0.90) = 3.85
Chapter 18: Statistical Quality Control
For R Chart:
Since n = 4, D3 = 0
D4 = 2.282
R = 0.90
Centerline:
UCL:
D4 R = (2.282)(0.90) = 2.05
LCL:
D3 R = 0
x Chart:
R Chart:
18.7
p̂1 = .025, p̂2 = .000, p̂3 = .025, p̂4 = .075,
p̂5 = .05, p̂6 = .125, p̂7 = .05
p = .050
Centerline:
p = .050
UCL: .05 + 3
(.05)(.95)
= .05 + .1034 = .1534
40
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LCL: .05 – 3
(.05)(.95)
= .05 – .1034 = .000
40
p Chart:
18.9
c =
43
= 1.34375
32
Centerline: c = 1.34375
UCL:
c  3 c = 1.34375 + 3 1.34375 =
1.34375 + 3.47761 = 4.82136
LCL:
c  3 c = 1.34375 – 3 1.34375 =
1.34375 – 3.47761 = 0.000
c Chart:
Chapter 18: Statistical Quality Control
339
18.11
While there are no points outside the limits, the first chart exhibits some problems. The chart ends
with 9 consecutive points below the centerline. Of these 9 consecutive points, there are at least 4
out of 5 in the outer 2/3 of the lower region. The second control chart contains no points outside
the control limit. However, near the end, there are 8 consecutive points above the centerline. The
p chart contains no points outside the upper control limit. Three times, the chart contains two out
of three points in the outer third. However, this occurs in the lower third where the proportion of
noncompliance items approaches zero and is probably not a problem to be concerned about.
Overall, this seems to display a process that is in control. One concern might be the wide swings
in the proportions at samples 15, 16 and 22 and 23.
18.13
n = 10
c=0
p0 = .05
P(x = 0) = 10C0(.05)0(.95)10 = .5987
1 – P(x = 0) = 1 – .5987 = .4013
The producer's risk is .4013
P(x = 0) = 15C0(.14)0(.86)10 =
p1 = .14
.2213
The consumer's risk is .2213
18.15
n=8
c=0
p
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
p0 = .03
Probability
.9227
.8506
.7837
.7214
.6634
.6096
.5596
.5132
.4703
.4305
.3937
.3596
.3282
.2992
.2725
p1 = .1
Producer's Risk for (p0 = .03) =
1 – .7837 = .2163
Consumer's Risk for (p1 = .10) = .4305
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OC Chart:
18.17
Stop

N

(no)
D  K  L  M (yes)  Stop


Stop


(no)
(no)
Start  A  B (yes) C  E  F  G
(yes)

H(no) J  Stop
(yes)



I     
Chapter 18: Statistical Quality Control
18.19
Fishbone Diagram:
Cause-and-Effect Diagram
E
C
A
Cause 1
Cause 1
Cause 1
Cause 2
Cause 2
Cause 2
Cause 3
Cause 3
Cause 5
Cause 4
Cause 4
Cause 3
Cause 3
Cause 2
Cause 2
Cause 1
Cause 1
Environment
18.21
D
B
p̂1 = .06, p̂2 = .22, p̂3 = .14, p̂4 = .04, p̂5 = .10,
p̂6 = .16, p̂7 = .00, p̂8 = .18, p̂9 = .02, p̂10 = .12
p=
52
= .104
500
Centerline:
p = .104
UCL: .104 + 3
(.104)(.896)
= .104 + .130 = .234
50
LCL: .104 – 3
(.104)(.896)
= .104 – .130 = .000
50
p Chart:
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18.23 n = 15, c = 0, p0 = .02, p1 = .10
p
.01
.02
.04
.06
.08
.10
.12
.14
Probability
.8601
.7386
.5421
.3953
.2863
.2059
.1470
.1041
Producer's Risk for (p0 = .02) = 1 – .7386 = .2614
Consumer's Risk for (p1 = .10) = .2059
OC Curve:
Chapter 18: Statistical Quality Control
18.25
x1 = 1.2100, x 2 = 1.2050, x 3 = 1.1900, x 4 = 1.1725,
x 5 = 1.2075, x 6 = 1.2025, x 7 = 1.1950, x 8 = 1.1950,
x 9 = 1.1850
R1 = .04, R2 = .02, R3 = .04, R4 = .04, R5 = .06, R6 = .02,
R7 = .07, R8 = .07, R9 = .06,
x = 1.19583
R = 0.04667
For x Chart:
Since n = 9, A2 = .337
x = 1.19583
Centerline:
x + A2 R = 1.19583 + .337(.04667) =
UCL:
1.19583 + .01573 = 1.21156
x – A2 R = 1.19583 – .337(.04667) =
LCL:
1.19583 – .01573 = 1.18010
For R Chart:
Centerline:
Since n = 9, D3 = .184
D4 = 1.816
R = .04667
UCL:
D4 R = (1.816)(.04667) = .08475
LCL:
D3 R = (.184)(.04667) = .00859
x Chart:
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R chart:
18.27
p̂1 = .12, p̂2 = .04, p̂3 = .00, p̂4 = .02667,
p̂5 = .09333, p̂6 = .18667, p̂7 = .14667, p̂8 = .10667,
p̂9 = .06667, p̂10 = .05333, p̂11 = .0000, p̂12 = .09333
p=
70
= .07778
900
Centerline:
p = .07778
UCL: .07778 + 3
(.07778)(.92222)
= .07778 + .09278 = .17056
75
.07778 – 3
(.07778)(.92222)
= .07778 – .09278 = .00000
75
LCL:
p Chart:
Chapter 18: Statistical Quality Control
18.29 n = 10
c=2
p
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
p0 = .10
p1 = .30
Probability
.9885
.9298
.8202
.6778
.5256
.3828
.2616
.1673
.0996
.0547
Producer's Risk for (p0 = .10) = 1 – .9298 = .0702
Consumer's Risk for (p1 = .30) = .3828
18.31
p̂1 = .05, p̂2 = .00, p̂3 = .15, p̂4 = .075,
p̂5 = .025, p̂6 = .025, p̂7 = .125, p̂8 = .00,
p̂9 = .10, p̂10 = .075, p̂11 = .05, p̂12 = .05,
p̂13 = .15, p̂14 = .025, p̂15 = .000
p=
36
= .06
600
Centerline:
p = .06
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UCL: .06 + 3
(.06)(.94)
= .06 + .11265 = .17265
40
LCL: .06 – 3
(.06)(.94)
= .06 – .112658 = .00000
40
p Chart:
18.33
There are some items to be concerned about with this chart. Only one sample range is above the
upper control limit. However, near the beginning of the chart there are eight sample ranges in a
row below the centerline. Later in the run, there are nine sample ranges in a row above the
centerline. The quality manager or operator might want to determine if there is some systematic
reason why there is a string of ranges below the centerline and, perhaps more importantly, why
there are a string of ranges above the centerline.
18.35
The centerline of the c chart indicates that the process is averaging 0.74 nonconformances per
part. Twenty-five of the fifty sampled items have zero nonconformances. None of the samples
exceed the upper control limit for nonconformances. However, the upper control limit is 3.321
nonconformances which, in and of itself, may be too many. Indeed, three of the fifty (6%)
samples actually had three nonconformances. An additional six samples (12%) had two
nonconformances. One matter of concern may be that there is a run of ten samples in which nine
of the samples exceed the centerline (samples 12 through 21). The question raised by this
phenomenon is whether or not there is a systematic flaw in the process that produces strings of
nonconforming items.