Download 18.Introduction to Quality

Statistics for
Business and Economics
6th Edition
Chapter 18
Introduction to Quality
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-1
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-2
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-3
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-4
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-5
Process Variation
Total Process
Common
Assignable
= Cause Variation + Cause Variation
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-6
Total Process Variation
Total Process
Common
Assignable
= Cause Variation + Cause Variation
Variation
Variation is often due to differences in:
 People
 Machines
 Materials
 Methods
 Measurement
 Environment
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-7
Common Cause Variation
Total Process
Common
Assignable
= Cause Variation + Cause Variation
Variation
Common cause variation
 naturally occurring and expected
 the result of normal variation in
materials, tools, machines, operators,
and the environment
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-8
Special Cause Variation
Total Process
Common
Assignable
= Cause Variation + Cause Variation
Variation
Special cause variation
 abnormal or unexpected variation
 has an assignable cause
 variation beyond what is considered
inherent to the process
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-9
Stable Process
 A process is stable (in-control) if
 all assignable causes are removed
 variation results only from common causes
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-10
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-11
Overview
Tools for Quality
Improvement
Control
Charts
Process
Capability
X-chart for the mean
s-chart for the standard deviation
P-chart for proportions
c-chart for number of occurrences
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-12
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-13
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
x   x i /K
i1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-14
Preliminaries
(continued)
 The sample standard deviations (denoted si for i = 1, 2,
. . . ,K) can be graphed on an s-chart
 The average sample standard deviation is
K
s   si /K
i1
 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-15
Example: Subgroups
 Sample measurements:
Subgroup Individual measurements
number
(subgroup size = 4)
Subgroup measures
Mean, x
Std. Dev., s
1
15
17
15
11
14.5
2.517
2
12
16
9
15
13.0
3.162
3
17
21
18
20
19.0
1.826
…
…
…
…
…
…
…
Average
subgroup
mean = x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Average
subgroup std.
dev. = s
Chap 18-16
Estimate of Process Standard
Deviation Based on s
 An estimate of process standard deviation is
σˆ  s/c 4
 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-17
Factors for Control Charts
 Selected control chart factors (Table 18.1)
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-18
Control Charts and Control Limits
 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)
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
UCL
+3σ
Process Average
- 3σ
LCL
time
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-19
Control Charts and Control Limits
(continued)
 The 3-standard-deviation control limits are estimated
for an X-chart as follows:
Process Average  3 Standard Deviations
 x  3 σˆ / n
 x  3 s /(c 4 n )
 x  A3s
Where the value of A 3 
3
is given in Table 18.1 or in Appendix 13
c4 n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-20
X-Chart
 The X-chart is a time plot of the sequence of
sample means
 The center line is
CL X  x
 The lower control limit is
LCL X  x  A 3 s
 The upper control limit is
UCL X  x  A 3 s
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-21
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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-22
X-Chart Example:
Subgroup Data
Day
1
2
3
4
5
6
7
Subgroup Subgroup Subgroup
Size
Mean
Std. Dev.
5
5
5
5
5
5
5
These are the xi values
for the 7 subgroups
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
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 si values
for the 7 subgroups
Chap 18-23
X-Chart
Control Limits Solution
x

x
i
K
s

s
5.32  6.59    6.79

 5.813
7
1.85  2.27    2.22

 2.151
K
7
i
UCL X  x  A 3 ( s )  5.813  (1.43)(2.1 51)  8.889
LCL X  x  A 3 ( s )  5.813  (1.43)(2.1 51)  2.737
A3 = 1.43 is from
Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-24
X-Chart
Control Chart Solution
Minutes
8
6
4
2
0
1
UCL = 8.889
_
x = 5.813
LCL = 2.737
2
3
4
Day
5
6
7
Conclusion: Process mean is in statistical control
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-25
s-Chart
 The s-chart is a time plot of the sequence of sample
standard deviations
 The center line on the s-chart is
CL  s
 The lower control limit (for three-standard error limits) is
LCLs  B3 s
 The upper control limit is
UCL s  B4 s
 Where the control chart constants B3 and B4 are found in Table 18.1 or
Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-26
s-Chart
Control Limits Solution
x

x
i
K
s

s
5.32  6.59    6.79

 5.813
7
1.85  2.27    2.22

 2.151
K
7
i
UCL s  B 4 s  (2.09)(2.1 51)  4.496
LCLs  B3 s  (0)(2.151)  0
B4 and B3 are found
in Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-27
s-Chart
Control Chart Solution
Minutes
UCL = 4.496
4
2
_
s = 2.151
0
LCL = 0
1
2
3
4
Day
5
6
7
Conclusion: Variation is in control
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-28
Control Chart Basics
Special Cause Variation:
Range of unexpected variability
UCL
Common Cause
Variation: range of
expected variability
+3σ
Process Average
- 3σ
LCL
time
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-29
Process Variability
Special Cause of Variation:
A measurement this far from the process average
is very unlikely if only expected variation is present
UCL
±3σ → 99.7% of
process values
should be in this
range
Process Average
LCL
time
UCL = Process Average + 3 Standard Deviations
LCL = Process Average – 3 Standard Deviations
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-30
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-31
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-32
Process In Control
 Process in control: points are randomly
distributed around the center line and all
points are within the control limits
UCL
Process Average
LCL
time
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-33
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-34
Process Not in Control
 One or more points outside
control limits
 Nine or more points in a row
on one side of the center line
UCL
UCL
Process
Average
Process
Average
LCL
LCL
 Six or more points moving in
the same direction
UCL
Process
Average
LCL
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-35
Out-of-control Processes
 When the control chart indicates an out-ofcontrol 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-36
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-37
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-38
Measures of Process Capability
Process capability is judged by the extent to which
x  3σˆ lies between the tolerance limits L and U
 Cp Capability Index
 Appropriate when the sample data are centered between the
tolerance limits, i.e. x  (L  U)/2
 The index is
UL
Cp 
6σˆ
 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)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-39
Measures of Process Capability
(continued)
 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
U  x x  L 
Cpk  Min
,

ˆ
ˆ
3σ 
 3σ
 A satisfactory value is at least 1.33
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-40
Process Capability
Example
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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-41
Process Capability:
Hotel Data
Day
Subgroup
Size
1
2
3
4
5
6
7
5
5
5
5
5
5
5
Subgroup
Mean
5.32
6.59
4.89
5.70
4.07
7.34
6.79
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Subgroup
Std. Dev.
1.85
2.27
1.28
1.99
2.61
2.84
2.22
Chap 18-42
Process Capability:
Hotel Example Solution
n5
X  5.813
s  2.151
c 4  0.940
s
2.151
Estimate σˆ 

 2.288
c 4 0.940
U  x x  L 
10  5.813 5.813  0 
Cpk  Min
,
 Min
,


ˆ
ˆ
3
σ
3
σ
3(2.228)
3(2.228)




 Min0.610 , 0.847   0.610
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-43
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”
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-44
p-Chart
(continued)
 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-45
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-46
p-Chart Example
Sample
Subgroup
number, i
Sample
size
Number of
successes
Proportion, pi
1
150
15
.1000
2
150
12
.0800
3
150
17
.1133
…
…
…
Average sample
proportions = p
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-47
Average of Sample Proportions
The average of sample proportions = p
If equal sample sizes:
K
p
p
i1
i
K
where:
pi = sample proportion for subgroup i
K = number of subgroups of size n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-48
Computing Control Limits
 The upper and lower control limits for a p-chart
are
UCL = Average Proportion + 3 Standard Deviations
LCL = Average Proportion – 3 Standard Deviations
 The standard deviation for the subgroup
proportions is
(
p
)(1

p
)
σˆ p 
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-49
Computing Control Limits
(continued)
 The upper and lower control limits for the
p-chart are
p(1  p)
LCLp  p  3
n
p(1  p)
UCLp  p  3
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Proportions are
never negative, so
if the calculated
lower control limit
is negative, set
LCL = 0
Chap 18-50
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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-51
p Chart Example:
Hotel Data
Day
1
2
3
4
5
6
7
# Rooms
200
200
200
200
200
200
200
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
# Not
Ready
16
7
21
17
25
19
16
Proportion
0.080
0.035
0.105
0.085
0.125
0.095
0.080
Chap 18-52
p Chart
Control Limits Solution
K
p
p
i1
K
i

.080  .035    .080
 .0864
7
LCLp  p  3
p(1  p)
.0864(1  .0864)
 .0864  3
 .0268
n
200
UCLp  p  3
p(1  p)
.0864(1  .0864)
 .0864  3
 .1460
n
200
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-53
p Chart
Control Chart Solution
P
0.15
UCL = .1460
_
p = .0864
0.10
0.05
0.00
LCL = .0268
1
2
3
4
5
Day
6
7
_
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-54
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-55
Mean and Standard Deviation
for a c-Chart
 The sample mean
number of occurrences is
c

c
i
K
 The standard deviation
for a c-chart is
σˆ c  c
where:
ci = number of successes per item
K = number of items sampled
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-56
c-Chart Center
and Control Limits
 The center line for a c-chart is
CLc  c
 The control limits for a c-chart are
LCLc  c  3 c
UCL c  c  3 c
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
The number of
occurrences can
never be negative,
so if the calculated
lower control limit
is negative, set
LCL = 0
Chap 18-57
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-58
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-59
Constructing the c-Chart
 The mean and standard deviation are:
c

c
K
i
2  1 3  0  5  1 0

 1.7143
7
c  1.7143  1.3093
 The control limits are:
UCL  c  3 c  1.7143  3(1.3093)  5.642
LCL  c  3 c  1.7143  3(1.3093)  2.214
Note: LCL < 0 so set LCL = 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-60
The completed c-Chart
6
UCL = 5.642
5
4
3
2
c = 1.714
1
0
LCL = 0
1
2
3
4
5
6
7
Sample number
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-61
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 18-62