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Quality Control.ppt
- 1. Chap 18-1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 18
Introduction to Quality
Statistics for
Business and Economics
6th Edition
- 2. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-2
Chapter Goals
After completing this chapter, you should be
able to:
Describe the importance of statistical quality control for
process improvement
Define common and assignable causes of variation
Explain process variability and the theory of control
charts
Construct and interpret control charts for the mean and
standard deviation
Obtain and explain measures of process capability
Construct and interpret control charts for number of
occurrences
- 3. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-3
The Importance of Quality
Primary focus is on process improvement
Data is needed to monitor the process and to insure the
process is stable with minimum variance
Most variation in a process is due to the system, not the
individual
Focus on prevention of errors, not detection
Identify and correct sources of variation
Higher quality costs less
Increased productivity
increased sales
higher profit
- 4. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-4
Variation
A system is a number of components that are
logically or physically linked to accomplish
some purpose
A process is a set of activities operating on a
system to transform inputs to outputs
From input to output, managers use statistical
tools to monitor and improve the process
Goal is to reduce process variation
- 5. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-5
Sources of Variation
Common causes of variation
also called random or uncontrollable causes of variation
causes that are random in occurrence and are inherent in all
processes
management, not the workers, are responsible for these causes
Assignable causes of variation
also called special causes of variation
the result of external sources outside the system
these causes can and must be detected, and corrective action
must be taken to remove them from the process
failing to do so will increase variation and lower quality
- 6. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-6
Process Variation
Variation is natural; inherent in the world
around us
No two products or service experiences
are exactly the same
With a fine enough gauge, all things can
be seen to differ
Total Process
Variation
Common
Cause Variation
Assignable
Cause Variation
= +
- 7. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-7
Total Process Variation
Total Process
Variation
Common
Cause Variation
Assignable
Cause Variation
= +
People
Machines
Materials
Methods
Measurement
Environment
Variation is often due to differences in:
- 8. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-8
Common Cause Variation
Common cause variation
naturally occurring and expected
the result of normal variation in
materials, tools, machines, operators,
and the environment
Total Process
Variation
Common
Cause Variation
Assignable
Cause Variation
= +
- 9. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-9
Special Cause Variation
Special cause variation
abnormal or unexpected variation
has an assignable cause
variation beyond what is considered
inherent to the process
Total Process
Variation
Common
Cause Variation
Assignable
Cause Variation
= +
- 10. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-10
Stable Process
A process is stable (in-control) if
all assignable causes are removed
variation results only from common causes
- 11. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-11
Control Charts
The behavior of a process can be monitored
over time
Sampling and statistical analysis are used
Control charts are used to monitor variation in a
measured value from a process
Control charts indicate when changes in data
are due to assignable or common causes
- 12. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-12
Overview
Process
Capability
Tools for Quality
Improvement
Control
Charts
X-chart for the mean
s-chart for the standard deviation
P-chart for proportions
c-chart for number of occurrences
- 13. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-13
X-chart and s-chart
Used for measured numeric data from a
process
Start with at least 20 subgroups of
observed values
Subgroups usually contain 3 to 6
observations each
For the process to be in control, both the
s-chart and the X-chart must be in control
- 14. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-14
Preliminaries
Consider K samples of n observations each
Data is collected over time from a measurable
characteristic of the output of a production process
The sample means (denoted xi for i = 1, 2, . . ., K) can
be graphed on an X-chart
The average of these sample means is the overall
mean of the sample observations
K
1
i
i/K
x
x
- 15. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-15
Preliminaries
The sample standard deviations (denoted si for i = 1, 2,
. . . ,K) can be graphed on an s-chart
The average sample standard deviation is
The process standard deviation, σ, is the standard
deviation of the population from which the samples
were drawn, and it must be estimated from sample data
/K
s
s
K
1
i
i
(continued)
- 16. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-16
Example: Subgroups
Sample measurements:
Subgroup measures
Subgroup
number
Individual measurements
(subgroup size = 4)
Mean, x Std. Dev., s
1
2
3
…
15
12
17
…
17
16
21
…
15
9
18
…
11
15
20
…
14.5
13.0
19.0
…
2.517
3.162
1.826
…
Average
subgroup
mean =
Average
subgroup std.
dev. = s
x
- 17. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-17
Estimate of Process Standard
Deviation Based on s
An estimate of process standard deviation is
Where s is the average sample standard deviation
c4 is a control chart factor which depends on the
sample size, n
Control chart factors are found in Table 18.1 or in
Appendix 13
If the population distribution is normal, this estimator
is unbiased
4
/c
s
σ
ˆ
- 18. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-18
Factors for Control Charts
n c4 A3 B3 B4
2 .789 2.66 0 3.27
3 .886 1.95 0 2.57
4 .921 1.63 0 2.27
5 .940 1.43 0 2.09
6 .952 1.29 0.03 1.97
7 .959 1.18 0.12 1.88
8 .965 1.10 0.18 1.82
9 .969 1.03 0.24 1.76
10 .973 0.98 0.28 1.72
Selected control chart factors (Table 18.1)
- 19. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-19
Process Average
Control Charts and Control Limits
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
UCL
LCL
+3σ
-3σ
time
A control chart is a time plot of the sequence of
sample outcomes
Included is a center line, an upper control limit (UCL)
and a lower control limit (LCL)
- 20. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-20
Control Charts and Control Limits
s
A
x
)
n
/(c
s
3
x
n
/
σ
3
x
Deviations
Standard
3
Average
Process
3
4
ˆ
The 3-standard-deviation control limits are estimated
for an X-chart as follows:
(continued)
Where the value of is given in Table 18.1 or in Appendix 13
n
c
3
A
4
3
- 21. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-21
X-Chart
The X-chart is a time plot of the sequence of
sample means
The center line is
The lower control limit is
The upper control limit is
s
A
x
LCL 3
X
x
CLX
s
A
x
UCL 3
X
- 22. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-22
X-Chart Example
You are the manager of a 500-room hotel.
You want to analyze the time it takes to deliver
luggage to the room. For seven days, you
collect data on five deliveries per day. Is the
process mean in control?
- 23. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-23
X-Chart Example:
Subgroup Data
Day Subgroup
Size
Subgroup
Mean
Subgroup
Std. Dev.
1
2
3
4
5
6
7
5
5
5
5
5
5
5
5.32
6.59
4.89
5.70
4.07
7.34
6.79
1.85
2.27
1.28
1.99
2.61
2.84
2.22
These are the xi values
for the 7 subgroups These are the si values
for the 7 subgroups
- 24. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-24
X-Chart
Control Limits Solution
5.813
7
6.79
6.59
5.32
K
x
x i
2.151
7
2.22
2.27
1.85
K
s
s i
2.737
51)
(1.43)(2.1
5.813
)
s
(
A
x
LCL
8.889
51)
(1.43)(2.1
5.813
)
s
(
A
x
UCL
3
X
3
X
A3 = 1.43 is from
Appendix 13
- 25. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-25
X-Chart
Control Chart Solution
UCL = 8.889
LCL = 2.737
0
2
4
6
8
1 2 3 4 5 6 7
Minutes
Day
x = 5.813
_
_
Conclusion: Process mean is in statistical control
- 26. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-26
s-Chart
The s-chart is a time plot of the sequence of sample
standard deviations
The center line on the s-chart is
The lower control limit (for three-standard error limits) is
The upper control limit is
Where the control chart constants B3 and B4 are found in Table 18.1 or
Appendix 13
s
B
LCL 3
s
s
CL
s
B
UCL 4
s
- 27. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-27
s-Chart
Control Limits Solution
5.813
7
6.79
6.59
5.32
K
x
x i
2.151
7
2.22
2.27
1.85
K
s
s i
0
(0)(2.151)
s
B
LCL
4.496
51)
(2.09)(2.1
s
B
UCL
3
s
4
s
B4 and B3 are found
in Appendix 13
- 28. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-28
s-Chart
Control Chart Solution
UCL = 4.496
0
2
4
1 2 3 4 5 6 7
Minutes
Day
LCL = 0
s = 2.151
_
Conclusion: Variation is in control
- 29. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-29
Process Average
Control Chart Basics
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
UCL
LCL
+3σ
-3σ
Common Cause
Variation: range of
expected variability
Special Cause Variation:
Range of unexpected variability
time
- 30. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-30
Process Average
Process Variability
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
UCL
LCL
±3σ → 99.7% of
process values
should be in this
range
time
Special Cause of Variation:
A measurement this far from the process average
is very unlikely if only expected variation is present
- 31. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-31
Using Control Charts
Control Charts are used to check for process
control
H0: The process is in control
i.e., variation is only due to common causes
H1: The process is out of control
i.e., assignable cause variation exists
If the process is found to be out of control,
steps should be taken to find and eliminate the
assignable causes of variation
- 32. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-32
In-control Process
A process is said to be in control when the
control chart does not indicate any out-of-control
condition
Contains only common causes of variation
If the common causes of variation is small, then
control chart can be used to monitor the process
If the variation due to common causes is too large,
you need to alter the process
- 33. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-33
Process In Control
Process in control: points are randomly
distributed around the center line and all
points are within the control limits
UCL
LCL
time
Process Average
- 34. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-34
Process Not in Control
Out of control conditions:
One or more points outside control limits
6 or more points in a row moving in the same
direction either increasing or decreasing
9 or more points in a row on the same side of
the center line
- 35. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-35
Process Not in Control
One or more points outside
control limits
UCL
LCL
Nine or more points in a row
on one side of the center line
UCL
LCL
Six or more points moving in
the same direction
UCL
LCL
Process
Average
Process
Average
Process
Average
- 36. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-36
Out-of-control Processes
When the control chart indicates an out-of-
control condition (a point outside the control
limits or exhibiting trend, for example)
Contains both common causes of variation and
assignable causes of variation
The assignable causes of variation must be identified
If detrimental to the quality, assignable causes of variation
must be removed
If increases quality, assignable causes must be incorporated
into the process design
- 37. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-37
Process Capability
Process capability is the ability of a process to
consistently meet specified customer-driven
requirements
Specification limits are set by management (in response
to customers’ expectations or process needs, for
example)
The upper tolerance limit (U) is the largest value that
can be obtained and still conform to customers’
expectations
The lower tolerance limit (L) is the smallest value that is
still conforming
- 38. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-38
Capability Indices
A process capability index is an aggregate
measure of a process’s ability to meet
specification limits
The larger the value, the more capable a
process is of meeting requirements
- 39. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-39
Measures of Process Capability
Process capability is judged by the extent to which
lies between the tolerance limits L and U
Cp Capability Index
Appropriate when the sample data are centered between the
tolerance limits, i.e.
The index is
A satisfactory value of this index is usually taken to be one that is at least
1.33 (i.e., the natural rate of tolerance of the process should be no more
than 75% of (U – L), the width of the range of acceptable values)
σ
3
x ˆ
σ
6
L
U
Cp
ˆ
U)/2
(L
x
- 40. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-40
Measures of Process Capability
Cpk Index
Used when the sample data are not centered between
the tolerance limits
Allows for the fact that the process is operating closer to
one tolerance limit than the other
The Cpk index is
A satisfactory value is at least 1.33
(continued)
σ
3
L
x
,
σ
3
x
U
Min
Cpk
ˆ
ˆ
- 41. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-41
You are the manager of a 500-room hotel.
You have instituted tolerance limits that
luggage deliveries should be completed
within ten minutes or less (U = 10, L = 0).
For seven days, you collect data on five
deliveries per day. You know from prior
analysis that the process is in control. Is the
process capable?
Process Capability
Example
- 42. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-42
Process Capability:
Hotel Data
Day Subgroup
Size
Subgroup
Mean
Subgroup
Std. Dev.
1
2
3
4
5
6
7
5
5
5
5
5
5
5
5.32
6.59
4.89
5.70
4.07
7.34
6.79
1.85
2.27
1.28
1.99
2.61
2.84
2.22
- 43. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-43
Process Capability:
Hotel Example Solution
0.610
0.847
,
0.610
Min
3(2.228)
0
5.813
,
3(2.228)
5.813
10
Min
σ
3
L
x
,
σ
3
x
U
Min
Cpk
ˆ
ˆ
0.940
c
2.151
s
5.813
X
5
n 4
2.288
0.940
2.151
c
s
σ
Estimate
4
ˆ
The capability index for the luggage delivery process is less than
1. The upper specification limit is less than 3 standard deviations
above the mean.
- 44. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-44
p-Chart
Control chart for proportions
Is an attribute chart
Shows proportion of defective or nonconforming
items
Example -- Computer chips: Count the number of
defective chips and divide by total chips inspected
Chip is either defective or not defective
Finding a defective chip can be classified a
“success”
- 45. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-45
p-Chart
Used with equal or unequal sample sizes
(subgroups) over time
Unequal sizes should not differ by more than ±25%
from average sample sizes
Easier to develop with equal sample sizes
Should have large sample size so that the
average number of nonconforming items per
sample is at least five or six
(continued)
- 46. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-46
Creating a p-Chart
Calculate subgroup proportions
Graph subgroup proportions
Compute average of subgroup proportions
Compute the upper and lower control limits
Add centerline and control limits to graph
- 47. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-47
p-Chart Example
Subgroup
number, i
Sample
size
Number of
successes
Sample
Proportion, pi
1
2
3
…
150
150
150
15
12
17
…
.1000
.0800
.1133
…
Average sample
proportions = p
- 48. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-48
Average of Sample Proportions
The average of sample proportions = p
where:
pi = sample proportion for subgroup i
K = number of subgroups of size n
If equal sample sizes:
K
p
p
K
1
i
i
- 49. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-49
Computing Control Limits
The upper and lower control limits for a p-chart
are
The standard deviation for the subgroup
proportions is
UCL = Average Proportion + 3 Standard Deviations
LCL = Average Proportion – 3 Standard Deviations
n
)
p
)(1
p
(
σp
ˆ
- 50. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-50
Computing Control Limits
The upper and lower control limits for the
p-chart are
(continued)
n
)
p
(1
p
3
p
UCL
n
)
p
(1
p
3
p
LCL
p
p
Proportions are
never negative, so
if the calculated
lower control limit
is negative, set
LCL = 0
- 51. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-51
p-Chart Example
You are the manager of a 500-room hotel.
You want to achieve the highest level of
service. For seven days, you collect data on
the readiness of 200 rooms. Is the process in
control?
- 52. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-52
p Chart Example:
Hotel Data
# Not
Day # Rooms Ready Proportion
1 200 16 0.080
2 200 7 0.035
3 200 21 0.105
4 200 17 0.085
5 200 25 0.125
6 200 19 0.095
7 200 16 0.080
- 53. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-53
p Chart
Control Limits Solution
.0864
7
.080
.035
.080
K
p
p
K
1
i
i
.1460
200
.0864)
.0864(1
3
.0864
n
)
p
(1
p
3
p
UCL
.0268
200
.0864)
.0864(1
3
.0864
n
)
p
(1
p
3
p
LCL
p
p
- 54. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-54
p = .0864
p Chart
Control Chart Solution
UCL = .1460
LCL = .0268
0.00
0.05
0.10
0.15
1 2 3 4 5 6 7
P
Day
Individual points are distributed around p without any pattern.
Any improvement in the process must come from reduction
of common-cause variation, which is the responsibility of
management.
_
_
- 55. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-55
c-Chart
Control chart for number of defects per item
Also a type of attribute chart
Shows total number of nonconforming items
per unit
examples: number of flaws per pane of glass
number of errors per page of code
Assume that the size of each sampling unit
remains constant
- 56. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-56
Mean and Standard Deviation
for a c-Chart
The sample mean
number of occurrences is
K
c
c i
The standard deviation
for a c-chart is
c
σc
ˆ
where:
ci = number of successes per item
K = number of items sampled
- 57. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-57
c-Chart Center
and Control Limits
c
3
c
UCL
c
3
c
LCL
c
c
The control limits for a c-chart are
c
CLc
The center line for a c-chart is
The number of
occurrences can
never be negative,
so if the calculated
lower control limit
is negative, set
LCL = 0
- 58. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-58
Process Control
Determine process control for p-chars and c-charts
using the same rules as for X and s-charts
Out of control conditions:
One or more points outside control limits
Six or more points moving in the same direction
Nine or more points in a row on one side of the center line
- 59. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-59
c-Chart Example
A weaving machine makes
cloth in a standard width.
Random samples of 10 meters
of cloth are examined for flaws.
Is the process in control?
Sample number 1 2 3 4 5 6 7
Flaws found 2 1 3 0 5 1 0
- 60. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-60
Constructing the c-Chart
The mean and standard deviation are:
1.7143
7
0
1
5
0
3
1
2
K
c
c i
1.3093
1.7143
c
2.214
3(1.3093)
1.7143
c
3
c
LCL
5.642
3(1.3093)
1.7143
c
3
c
UCL
The control limits are:
Note: LCL < 0 so set LCL = 0
- 61. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-61
The completed c-Chart
The process is in control. Individual points are distributed around
the center line without any pattern. Any improvement in the
process must come from reduction in common-cause variation
UCL = 5.642
LCL = 0
Sample number
1 2 3 4 5 6 7
c = 1.714
6
5
4
3
2
1
0
- 62. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 18-62
Chapter Summary
Reviewed the concept of statistical quality
control
Discussed the theory of control charts
Common cause variation vs. special cause variation
Constructed and interpreted X and s-charts
Obtained and interpreted process capability
measures
Constructed and interpreted p-charts
Constructed and interpreted c-charts