Six Sigma Quality and
Statistical Process Control
Chapter 7
1
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Definition:
Total Quality Management
• Total Quality Management (TQ, QM or
TQM) and Six Sigma (6) are sweeping
“culture change” efforts to position a
company for greater customer
satisfaction, profitability and
competitiveness.
• TQ may be defined as managing the entire
organization so that it excels on all
dimensions of products and services that
are important to the customer.
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Total Quality Is…
• Meeting Our Customer’s Requirements
• Doing Things Right the First Time; Freedom
from Failure (Defects)
• Consistency (Reduction in Variation)
• Continuous Improvement
• Quality in Everything We Do
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The Continuous
Improvement Process
Empowerment/
Shared Leadership
Customer
Satisfaction
Business
Results
Team
Management
Process
Improvement/
Problem
Solving
Measurement
Measurement
Measurement
...
Measurement
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4
Is 99% Quality Good
Enough?
• 22,000 checks will be deducted from
the wrong bank accounts in the next
60 minutes.
• 20,000 incorrect drug prescriptions
will be written in the next 12 months.
• 12 babies will be given to the wrong
parents each day.
5
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Defects Per Million Opportunities (DPMO)
·
100K
But is Six Sigma Realistic?
·
IRS – Tax Advice (phone-in)
(66810 ppm)
10K
41
Average
Company
1K
31
····
···
Restaurant Bills
Doctor Prescription Writing
Payroll Processing
Order Write-up
Journal Vouchers
Wire Transfers
Air Line Baggage Handling
Purchased Material
Lot Reject Rate
100
21
·
(233 ppm)
10
11
Best in Class
1
1
Domestic Airline
Flight Fatality Rate
(3.4 ppm)
2
3
3
4
4
SIGMA
5
5
6
6
7
(0.43 ppm)
7
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6
Six Sigma Quality
The objective of Six Sigma quality is 3.4
defects per million opportunities!
sigma level
1
2
3
4
5
6
6 with 1.5 stdev
mean shift
probability
0.84
0.977
0.9987
0.99997
0.9999997
0.99999999901
0.999996599
this is the amount
that exists to the left
OR the right of each total amount
tail (1-pr)
under the curve
0.158655259759
68.2689480%
0.022750062036
95.4499876%
0.001349967223
99.7300066%
0.000031686035
99.9936628%
0.000000287105
99.9999426%
0.000000000990
99.9999998%
0.000003400803
error rate
32 out of 100
4.6 out of 100
3 out of 1000
7 out of 100,000
6 out of 10,000,000
2 out of 1,000,000,000 (a billion)
99.9996599% 3.4 out of 1,000,000
7
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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
8
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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
9
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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
10
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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
11
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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
12
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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
13
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Control Charts
 Graph establishing process control
limits
 Charts for variables
Mean (x-bar), Range (R)
 Chart for attributes
P Chart
C Chart
14
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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
Sample number
7
8
9
10
15
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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
16
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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
17
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Control Chart for
Attributes
 p Charts
Calculate percent defectives in sample
18
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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
19
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The Normal Distribution
95%
99.74%
-3
-2
-1
=0
1
2
3
20
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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
21
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
Example 15.1
NUMBER OF
DEFECTIVES
PROPORTION
DEFECTIVE
6
0
4
:
:
18
200
.06
.00
.04
:
:
.18
22
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
Example 15.1
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
23
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p-Chart Example
20 samples of 100 pairs of jeans
SAMPLE
1
2
3
:
:
20
Example 15.1
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
24
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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
25
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C Chart
• Used when you can’t calculate a
proportion defective and an actual
count is used.
• Key –the number of defects is
assumed to come from a large
population
• Ex. Defects in the paint job of a car
26
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C Chart con’t
• The mean is the average counted
number of defects per item (total
divided number of samples
• The sample standard deviation is
√cbar (square root of the mean of C)
27
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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
28
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Range ( R- ) Chart
UCL = D4R
LCL = D3R
R
R= k
where
R = range of each sample
k = number of samples
29
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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
Table 15.1
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
30
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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
31
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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
Example 15.3
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
32
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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
33
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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
34
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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
35
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Using x- and R-Charts
Together
 Each measures the process
differently
 Both process average and variability
must be in control
36
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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
37
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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
39
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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
40
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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
41
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Process Capability
Measures
Process Capability Index
Cpk = minimum
=
x - lower specification limit
,
3
=
upper specification limit - x
3
42
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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
43
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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