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```COMPLETE
BUSINESS
STATISTICS
by
AMIR D. ACZEL
&
JAYAVEL SOUNDERPANDIAN
7th edition.
Prepared by Lloyd Jaisingh, Morehead State
University
Chapter 13
Quality Control and Improvement
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
13-2
13 Quality Control and Improvement
• Using Statistics
• W. Edwards Deming Instructs
• Statistics and Quality
• The x-bar Chart
• The R Chart and the s Chart
• The p Chart
• The c Chart
• The x Chart
13-3
13 LEARNING OBJECTIVES
After studying this chapter you will be able to:
• Determine when to use control charts
• Create control charts for sample means, ranges and standard deviations
• Create control charts for sample proportions
• Create control charts for the number of defectives
• Draw Pareto charts using spreadsheet templates
• Draw control charts using spreadsheet templates
13-4
13-3 Statistics and Quality
A control chart is a time plot of a statistic, such as a sample mean, range,
standard deviation, or proportion, with a center line and upper and lower
control limits. The limits give the desired range of values for the statistic.
When the statistic is outside the bounds, or when its time plot reveals certain
patterns, the process may be out of control.
Value
This point is out of the control limits
UCL
3
3
Center
Line
LCL
Time
A process is considered in statistical control if it has no assignable causes,
only natural variation.
13-5
Control Charts
Value
Process is in control
Time
Value
Process mean varies
over time: process is
out of control
Time
13-6
Control Charts (Continued)
Value
Process variance
changes over time:
process is out of
control.
Process mean and
variance change
over time: process is
out of control.
Time
Value
Time
13-7
Pareto Diagrams – Using the Template
A Pareto diagram is a bar chart of the various problems in
production and their percentages, which must add to 100%.
A Pareto chart helps
to identify the most
significant problems
and thus one can
concentrate on their
solutions rather than
waste time and
resources on
unimportant causes.
13-8
Acceptance Sampling
•Finished products are grouped in lots before being shipped to
customers.
•The lots are numbered, and random samples from these lots
are inspected for quality.
•Such checks are made before the lots are shipped and after the
lots arrive at their destination.
•The random samples are measured to find out which and how
many items do not meet specifications
•A lot is rejected whenever the sample mean exceeds or falls
below some pre-specified limit.
13-9
Acceptance Sampling
• For attribute data, the lot is rejected when the
number of defectives or non-conforming items in
the sample exceeds a pre-specified limit.
• Acceptance sampling does not improve quality by
itself.
• It simply removes bad lots.
• To improve quality, it is necessary to control the
production process itself, removing any assignable
causes and striving to reduce the variation in the
process.
13-10
Six Sigma
• Six Sigma is a further innovation, beyond Deming’s
work, in the field of quality assurance and control.
• The purpose of Six Sigma is to push defect levels
below a certain specified threshold.
• Six Sigma helps to improve quality.
• The key to Six Sigma is a precise definition of the
production process with accurate measurements and
valid collection of data.
13-11
Six Sigma
• It also involves detailed analysis to measure the
relationships and causality of key factors in the
production process.
• Experimental Design is used to identify these key
factors.
• Strict control of the production process is exercised.
Any variations are corrected, and the process is
further monitored as it goes on line.
• The essence of Six Sigma is the statistical methods
described in this chapter.
13-12
13-4 The X-Bar Chart: A Control Chart
for the Process Mean
Elements of a control chart for the process mean:
k
 xi
Center line: x  i 1
k
LCL: x  A2 R
UCL: x  A2 R
where: k = number of samples, each of size n
xi = Sample mean for sample i
R  Range of sample i
i
k
R
i
i=1
R =
k
If the sample size in each group is more than 10:
s / c4
s / c4
LCL = x - 3
UCL = x + 3
n
n
where s is the average of the standard deviations of all groups.
n
A2
2
3
4
5
6
7
8
9
10
15
20
25
1.880
1.023
0.729
0.577
0.483
0.419
0.373
0.337
0.308
0.223
0.180
0.153
c4
0.7979
0.8862
0.9213
0.9400
0.9515
0.9594
0.9650
0.9693
0.9727
0.9823
0.9869
0.9896
13-13
The X-Bar Chart: A Control Chart for
the Process Mean (Continued)
• Tests for assignable causes:
One point beyond 3 (3s)
Nine points in a row on one side of the center line
Six points in a row steadily increasing or decreasing
Fourteen points in a row alternating up and down
Two out of three points in a row beyond 2 (2s)
Four out of five points in a row beyond 1 (1s)
Fifteen points in a row within 1 (1s) of the center line
Eight points in a row on both sides of the center line, all beyond 1 (1s)
13-14
Tests for Assignable Causes
Value
3
2
1
1
2
3

Test 1: One value beyond
3 (3s)
Time
Value
3
2
1
1
2
3

Test 2: Nine points in a
row on one side
of the center line.
Time
13-15
Tests for Assignable Causes
(Continued)
Value
3
2
1
1
2
3

Test 3: Six points in a
row steadily
increasing or
decreasing.
Time
Value
3
2
1
1
2
3

Test 4: Fourteen points in
a row alternating
up and down.
Time
13-16
Tests for Assignable Causes
(Continued)
Value
3
2
1
1
2
3

Test 5: Two out of three
points in a row
beyond 2 (2s)
Time
Value
3
2
1
1
2
3

Test 6: Four out of five
points in a row
beyond 1 (1s)
Time
13-17
Tests for Assignable Causes
(Continued)
Value
3
2
1
1
2
3

Test 7: Fifteen points in a
row within 1
(1s) of the center
line.
Time
Value
3
2
1
1
2
3

Test 8: Eight points in a
row on both sides
of the center line,
all beyond 1 (1s)
Time
13-18
X-bar Chart: Example 13-1 – Using
the Template
13-19
X-bar Chart: Example 13-1(continued)
– Using the Template
Note: The X-bar chart cannot be interpreted unless the
R or s chart has been examined and is in control.
13-20
X-bar Chart: Example 13-1(continued)
– Using Minitab
Xbar Chart of Concentration
10.8
UCL=10.784
Sample Mean
10.6
10.4
_
_
X=10.257
10.2
10.0
9.8
LCL=9.731
9.6
1
2
3
4
5
6
Sample
7
8
9
10
Note: The X-bar chart cannot be interpreted unless the
R or s chart has been examined and is in control.
13-21
13-5 The R Chart and s Chart
Elements of a control chart for the process range:
Center line: R
LCL: D3 R
UCL: D4 R
k
R
i
i=1
where: R =
k
Elements of a control chart for the process standard
deviation:
Center line: s
LCL: B3 s
UCL: B4 s
k
s
i
i=1
where: s =
k
n
2
3
4
5
6
7
8
9
10
15
20
25
D3
0
0
0
0
0
0.076
0.136
0.184
0.223
0.348
0.414
0.459
D4
3.267
2.575
2.282
2.115
2.004
1.924
1.864
1.816
1.777
1.652
1.586
1.541
B3
0
0
0
0
0.030
0.118
0.185
0.239
0.284
0.428
0.510
0.565
B4
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
1.572
1.490
1.435
13-22
R Chart: Example 13-1 using the
Template
The process range seems to be in control.
13-23
s Chart: Example 13-1 using the
Template
The process standard deviation seems to
be in control.
13-24
Example 13-2 using the Template
13-25
Example 13-2 using the Template Continued
13-26
Example 13-2 using the Template Continued
Based on the x-bar, R, and s charts, the process
seems to be in control.
13-27
Example 13-2 using Minitab
Xbar Chart of Delivery Times
UCL=7.196
7
Sample Mean
6
_
_
X=4.8
5
4
3
LCL=2.404
2
1
2
3
4
5
6
Sample
7
8
9
10
13-28
Example 13-2 using Minitab
R Chart of Delivery Times
UCL=6.029
6
Sample Range
5
4
3
_
R=2.342
2
1
0
LCL=0
1
2
3
4
5
6
Sample
7
8
9
10
13-29
Example 13-2 using Minitab
S Chart of Delivery Times
3.5
UCL=3.149
3.0
Sample StDev
2.5
2.0
1.5
_
S=1.226
1.0
0.5
0.0
LCL=0
1
2
3
4
5
6
Sample
7
8
9
10
Based on the x-bar, R, and s charts, the process
seems to be in control.
13-30
13-6 The p Chart: Proportion of
Defective Items
Elements of a control chart for the process
proportion:
Center line: p
p(1 - p)
p(1 - p)
LCL: p - 3
UCL: p + 3
n
n
where: n is the number of elements in each
sample
p is the proportion of defectives in
the combined, overall sample
13-31
13-6 The p Chart: Proportion of Defective Items
– Using the Template for Example 13-3
Process is out of control – Two points fall outside the control limit
13-32
13-6 The p Chart: Proportion of Defective
Items – Using Minitab for Example 13-3
P Chart of Defectives
0.4
1
1
0.3
Proportion
UCL=0.2624
0.2
_
P=0.1125
0.1
0.0
LCL=0
1
2
3
4
5
6
7
Sample
8
9
10
11
12
Process is out of control – Two points fall outside the control limit
13-33
13-7 The c Chart: (Defects Per Item)
Elements of a control chart for the number
of imperfections per item, c:
Center line: c
LCL: c - 3 c
UCL: c + 3 c
where: c is the average number of defects
or imperfections per item (or area,
volume, etc. )
13-34
The c Chart: Example 13-4 using the
Template
Observe that one observation is outside the upper control limit,
indicating that the process may be out of control. The general
downward trend should be investigated.
13-35
The c Chart: Example 13-4 using
Minitab
C Chart of Nonconformaties
18
1
16
UCL=15.81
Sample Count
14
12
10
_
C=7.56
8
6
4
2
0
LCL=0
1
3
5
7
9
11
13
15
Sample
17
19
21
23
25
Observe that one observation is outside the upper control limit,
indicating that the process may be out of control. The general
downward trend should be investigated.
13-36
13-8 The x Chart
Sometimes we are interested in
controlling the process mean, but
our observations come so slowly
from the production process that
we cannot aggregate them into
groups. In such case we may
consider an x chart. An x-chart
is a chart for the raw values of
the variable in question.
The chart is
effective if the
variable has an
approximate
normal
distribution. The
bounds are  3
standard
deviations from
the mean of the
process.
13-37
13-8 The x Chart for Example 13-3 –
Using Minitab
I Chart of Defectives
20
UCL=17.80
Individual Value
15
10
_
X=4.5
5
0
-5
LCL=-8.80
-10
1
2
3
4
5
6
7
8
Observation
9
10
11
12
NOTE: The X-Chart
Is same as the
Individual chart in
Minitab
13-38
13-8 The x Chart for Example 13-4 –
Using Minitab
I Chart of Nonconformaties
20
UCL=18.75
Individual Value
15
10
_
X=7.56
5
0
LCL=-3.63
-5
1
3
5
7
9
11 13
15
Observation
17
19
21
23
25
```