business statistics, 4e, by ken black. © 2003 john wiley & sons. 18-1 business statistics, 4e...
TRANSCRIPT
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-1
Business Statistics, 4eby Ken Black
Chapter 18
Statistical Quality Control
Discrete Distributions
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-2
• 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 chars.
• Learn how to construct charts, R charts, P charts, and c charts.
• Understand the theory and application of acceptance sampling.
Learning Objectives
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-3
Quality
• Quality is when a product delivers what is stipulated for in its specifications
• Crosby: “quality is conformance to requirements”
• Feigenbaum: “quality is a customer determination”
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-4
Garvin’s Five Dimensions of Quality
• Transcendent quality: “innate excellence”• Product quality: quality is measurable• User quality: quality is determined by the
consumer• Manufacturing quality: quality is measured
by the manufacturer's ability to target the product specifications with little variability
• Value Quality: did the consumer get his or her money’s worth?
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-5
Quality Control• Quality control is the collection of strategies,
techniques, and actions taken by an organization to assure themselves that they are producing a quality product.
• After-process quality control involves 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.– reporting of the number of defects per time period– screening defective products from consumers
• In-process quality control techniques measure product attributes at various intervals throughout the manufacturing process in an effort to pinpoint problem areas.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-6
Deming’s Fourteen Points1.Create constancy of purpose fro improvement of product and service.2.Adopt a new philosophy.3.Cease dependence on mass inspection.4.End the practice of awarding business on price tag alone.5.Improve constantly and forever the system of production and service.6.Institute training.7.Institute leadership.8Drive out fear.9.Break down barriers between staff areas.10.Eliminate slogans.11.Eliminate numerical quotas.12.Remove barriers to pride of workmanship.13.Institute a vigorous program of education and retraining.14.Take action to accomplish the transformation.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-7
Important Quality Concepts• Benchmarking
– examine and emulate the best practices and techniques used in the industry
– a positive, proactive process to make changes that will effect superior performance
• Just-In-Time Inventory Systems– necessary parts for production arrive “just in time”– reduced holding costs, personnel, and space needed for inventory
• Reengineering– complete redesign of the core business process in a company
• Six sigma– Total quality approach that measures the capacity of a process to
perform defect -free work• Team Building:
– employee groups take on managerial responsibilities– quality circle
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-8
Process Analysis• A process is a series of actions, changes or functions
that bring about a result.• Flowcharts - schematic representation of all the
activities and interactions that occur in a process• Pareto Analysis -quantitative tallying of the number and
types of defects that occur with a product• Pareto Chart - ranked vertical bar chart with most
frequently occurring on the left• Fishbone Diagram - display of potential cause-and-
effect relationships• Control Charts - graphical method for evaluating
whether a process is or is not in a “state of statistical control”
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-9
Flowchart Symbols
Input/Output Symbol
Processing Symbol
Decision Symbol
Start/Stop Symbol
Flow line Symbol
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-10
Pareto ChartPareto Chart
0
10
20
30
40
50
60
70
80
90
100
PoorWiring
Short inCoil
DefectivePlug
Other
Fre
qu
ency
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-11
Cause-and-Effect Diagram
Raw MaterialsEquipment
WorkersMethodology
PoorWiring
Wiring Scheme
Pland Layout
MaintenanceTools
Out-of-AdjustmentOut-of-Date
Vendor
Transportation
Inventory
Training
Attitude
Absenteeism
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-12
Control Chart
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sample Number
Sample Mean
UCL
LCL
Centerline
XX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-13
Types of Control Charts
• Control charts for measurements– charts– R charts
• Control charts for compliance items– P charts– c charts
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-14
Control Chart
Monitor process location (center)1.Decide on the quality to be measured.2.Determine a sample size.3.Gather 20 to 30 samples.4.Compute the sample average for each sample.5.Compute the sample range for each sample.6.Determine the average sample mean for all samples.7.Determine the average sample range (or sample standard deviation) for all samples.8Using the size of the samples, determine the value of A2 or A3.9.Compute the UCL and the LCL
XX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-15
Control Chart: Formulas
X and Charts
XX
k
UCL X R
LCL X R
RR
k
LCL R
UCL R
AA
DD
R
2
2
3
4
X and S Charts
XX
k
UCL X R
LCL X R
SS
k
UCL R
LCL R
AA
BB
3
3
4
3
XX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-16
Data for Demonstration Problem 18.1: Samples 1 - 10
1 2 3 4 5 6 7 8 9 105.13 4.96 5.21 5.02 5.12 4.98 4.99 4.96 4.96 5.034.92 4.98 4.87 5.09 5.08 5.02 5.00 5.01 5.00 4.995.01 4.95 5.02 4.99 5.09 4.97 5.00 5.02 4.91 4.964.88 4.96 5.08 5.02 5.13 4.99 5.02 5.05 4.87 5.145.05 5.01 5.12 5.03 5.06 4.98 5.01 5.04 4.96 5.114.97 4.89 5.04 5.01 5.13 4.99 5.01 5.02 5.01 5.04
4.9933 4.9583 5.0567 5.0267 5.1017 4.9883 5.0050 5.0167 4.9517 5.04500.25 0.12 0.34 0.10 0.07 0.05 0.03 0.09 0.14 0.18
XR
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-17
Data for Demonstration Problem 18.1: Samples 11 - 20
11 12 13 14 15 16 17 18 19 204.91 4.97 5.09 4.96 4.99 5.01 5.05 4.96 4.90 5.044.93 4.91 4.96 4.99 4.97 5.04 4.97 4.93 4.85 5.035.04 5.02 5.05 4.82 5.01 5.09 5.04 4.97 5.02 4.975.00 4.93 5.12 5.03 4.98 5.07 5.03 5.01 5.01 4.994.90 4.95 5.06 5.00 4.96 5.12 5.09 4.98 4.88 5.054.82 4.96 5.01 4.96 5.02 5.13 5.01 4.92 4.86 5.06
4.9333 4.9567 5.0483 4.9600 4.9883 5.0767 5.0317 4.9617 4.9200 5.02330.22 0.11 0.16 0.21 0.06 0.12 0.12 0.09 0.17 0.09
XR
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-18
Demonstration Problem 18.1:Control Chart Computations
X and Charts
XX
k
UCL X R
LCL X R
RR
k
LCL R
UCL R
AA
DD
R
4 9933 4 9583 5 0566 5 0233
205 00215
5 00215 0 483 0136 5 06784
5 00215 0 483 0136 4 93646
0 25 012 0 34 0 09
200136
0 0136 0
2 004 0136 0 2725
2
2
3
4
. . . ..
. . . .
. . . .
. . . ..
.
. . .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-19
Sigma level: 3
2019
1817
1615
1413
1211
109
87
65
43
21
Bearing Diameter
UCL = 5.0679
Average = 5.0022
LCL = 4.9364
Control Chart: Bearing Diameter
Mean
5.10963
5.05590
5.00217
4.94844
4.89471
Demonstration Problem 18.1: Control ChartXX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-20
R ChartMonitor process Monitor process variationvariation
1.1.Decide on the quality to be measured.Decide on the quality to be measured.2.2.Determine a sample size.Determine a sample size.3.3.Gather 20 to 30 samples.Gather 20 to 30 samples.4.4.Compute the sample range for each sample.Compute the sample range for each sample.5.5.Determine the average sample mean for all Determine the average sample mean for all samples.samples.6.6.Using the size of the samples, determine the Using the size of the samples, determine the values of Dvalues of D33 and D and D44..7.7.Compute the UCL and the LCLCompute the UCL and the LCL
Monitor process Monitor process variationvariation1.1.Decide on the quality to be measured.Decide on the quality to be measured.2.2.Determine a sample size.Determine a sample size.3.3.Gather 20 to 30 samples.Gather 20 to 30 samples.4.4.Compute the sample range for each sample.Compute the sample range for each sample.5.5.Determine the average sample mean for all Determine the average sample mean for all samples.samples.6.6.Using the size of the samples, determine the Using the size of the samples, determine the values of Dvalues of D33 and D and D44..7.7.Compute the UCL and the LCLCompute the UCL and the LCL
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-21
R Chart Formulas
R Charts
RR
k
LCL R
UCL R
DD
3
4
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-22
Demonstration Problem 18.2: R Control Chart
Control Chart: Bearing Diameter
Sigma level: 3
2019
1817
1615
1413
1211
109
87
65
43
21
Range
.4
.3
.2
.1
0.0
Bearing Diameter
UCL = .2725
Average = .1360
LCL = .0000
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-23
P ChartsMonitor Monitor proportion in noncomplianceproportion in noncompliance
1.1.Decide on the quality to be measured.Decide on the quality to be measured.2.2.Determine a sample size.Determine a sample size.3.3.Gather 20 to 30 samples.Gather 20 to 30 samples.4.4.Compute the sample proportion for each Compute the sample proportion for each sample.sample.5.5.Determine the average sample proportion Determine the average sample proportion for all samples.for all samples.6.6.Compute the UCL and the LCLCompute the UCL and the LCL
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-24
P Chart Formulas
:
:
:
pn
the number of noncomplying items in the sample
the number of items in the sample
=
the sample proportion
the number of samples
UCL = P + 3P Q
n
LCL P 3P Q
n
non
non
n
n
where
n
Pp
kwhere p
k
where Q P1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-25
Demonstration Problem 18.3: Twenty Samples of Bond Paper
Sample n
Number Out of
Compliance Sample n
Number Out of
Compliance1 50 4 11 50 22 50 3 12 50 63 50 1 13 50 04 50 0 14 50 25 50 5 15 50 16 50 2 16 50 67 50 3 17 50 28 50 1 18 50 39 50 4 19 50 1
10 50 2 20 50 5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-26
Demonstration Problem 18.3: Preliminary Calculations
Sample n nnon Sample n nnon
1 50 4 0.08 11 50 2 0.042 50 3 0.06 12 50 6 0.123 50 1 0.02 13 50 0 0.004 50 0 0.00 14 50 2 0.045 50 5 0.10 15 50 1 0.026 50 2 0.04 16 50 6 0.127 50 3 0.06 17 50 2 0.048 50 1 0.02 18 50 3 0.069 50 4 0.08 19 50 1 0.02
10 50 2 0.04 20 50 5 0.10
pp
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-27
Demonstration Problem 17.3:Centerline, UCL, and LCL Computations
Pp
k
UCL PP Q
n
LCL PP Q
nLCL
. . . ..
.. .
.
.. .
.
08 06 02 10
20053
3 053 3053 947
50148
3 053 3053 947
50042
0
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-28
Demonstration Problem 17.3:P Control Chart
0.000.020.040.060.080.100.120.140.16
0 5 10 15 20
Sample Number
P = .053
UCL = .148
LCL = 0
p
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-29
c ChartsMonitor number of nonconformances per item
1.Decide on nonconformances to be evaluated.2.Determine the number of items to be studied (at least 25).3.Gather items.4.Determine the value of c for each item by summing the number of nonconformances in the item.5.Determine the average number of nonconformances per item.6.Determine the UCL and the LCL
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-30
c Chart Formulas
ci
where
UCL c c
LCL c c
c c c ci
1 2 3
3
3
: i = number of items
number of nonconformities per item
= +
-
ic
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-31
Demonstration Problem 18.4:Number of Nonconformities in Oil Gauges
Item Number
Number of Nonconformities
Item Number
Number of Nonconformities
1 2 14 22 0 15 13 3 16 44 1 17 05 2 18 26 5 19 37 3 20 28 2 21 19 0 22 3
10 0 23 211 4 24 012 3 25 313 2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-32
Demonstration Problem 18.4:c Chart Calculations
ci
UCL c c
LCL c c
LCL
c c c ci
1 2 3 2 0 3 3
252 0
3 2 0 3 2 0 6 2
3 2 0 3 2 0 2 2
0
.
. . .
. . .
= +
-
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-33
Demonstration Problem 18.4: c Chart
01234567
0 5 10 15 20 25Item Number
c
UCL = 6.2
LCL = 0
c = 2.0
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-34
Interpreting Control Charts• Points are above UCL and/or below LCL• Eight or more consecutive points fall above or
below the centerline. Ten out of 11 points fall above or below the centerline. Twelve out of 14 points fall above or below the centerline.
• A trend of 6 or more consecutive points (increasing or decreasing) is present
• Two out of 3 consecutive values are in the outer one-third.
• Four out 5 consecutive values are in the outer two-thirds.
• The centerline shifts from chart to chart.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-35
Interpreting Control Charts:Points above UCL and/or below LCL
UCL
LCL
Centerline
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-36
Interpreting Control Charts: 8 Consecutive Points on One Side of the Centerline
UCL
LCL
Centerline
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-37
Interpreting Control Charts:7 Consecutive Increasing Points
UCL
LCL
Centerline
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-38
Interpreting Control Charts:2 out of 3 Consecutive Points in Outer 1/3
UCL
LCL
Centerline
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-39
Interpreting Control Charts:4 out of 5 Consecutive Points in Outer 2/3
UCL
LCL
Centerline
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-40
Acceptance Sampling
• Acceptance sampling is the inspection of a sample from a lot of goods to determine if the lot will be accepted or rejected.– N = the lot size– n = the sample size
• Single Sample Plan• Double-Sample Plan• Multiple-Sample Plan
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-41
Rules for Sampling Plans
Accept lot if
Reject lot if
x c
x c
First sample:Accept if
Reject if
Take second sample if
Second sample:Accept if
Reject if
1
1
1
1
1
xx
cxx
1
1
1 1
2 2
2 2
cr
x rx cx c
Single SamplePlan
Double SamplePlan
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-42
Producer’s and Consumer’s Risk
State of Nature
Null True Null False
Actions
Fail toReject Null
Type II errorCorrectDecision --
Consumer’sRisk
Reject NullType Ierror --Producer’sRisk
CorrectDecision
H0: the lot is of acceptable quality
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-43
Bicycle Manufacturer Example• N = 3,000 (Braces arrive at the manufacturer’s plant in lots of 3,000.)• n = 15 (The bicycle manufacturer randomly selects a sample of 15 braces
to inspect.)• X is the number of nonconforming braces in the
sample of 15.• A 2% nonconformance rate is acceptable to the
consumer (the bicycle manufacturer).• If the lot contains 60 nonconforming braces, what is
the probability that the consumer will reject the lot (producer’s risk)?
• If the lot contains 360 nonconforming braces, what is the probability that the consumer will not reject the lot (consumer’s risk)?
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Bicycle Manufacturer Example:Sampling Plan
n
c
x
x
15
1
1
1
=
Accept lot if
Reject lot if
n
c
x
x
15
1
1
1
=
Accept lot if
Reject lot if
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Bicycle Manufacturer Example:Analysis for 2% Nonconforming Braces
0
0 15 1 14
02
0 115
0
15
1
1 9647 0353
02 98 02 98
p
P x P x
.
.
. .
. . . .
Probability of accepting
Probability of rejecting
9647
Producer’s RiskProducer’s Risk
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Bicycle Manufacturer Example:Analysis for 12% Nonconforming
Braces
1
0 15 1 14
12
0 115
0
15
14476
1 4476 5524
12 88 12 88
p
P x P x
.
.
. .
. . . .
Probability of accepting
Probability of rejecting
Consumer’sRisk
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-47
Bicycle Manufacturer Example:OC Curve for n = 15 and c = 1
0.00
0.20
0.40
0.60
0.80
1.00
0% 10% 20% 30% 40%
Percent nonconforming
Probability of acceptance
2%
.9647 } .0353 Producer’s Risk
.4476 Consumer’s Risk
12%
.4476
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-48
Bicycle Manufacturer Example:OC Curve for n = 15 and c = 0
0.00
0.20
0.40
0.60
0.80
1.00
0% 10% 20% 30% 40%
Percent nonconforming
Probability of acceptance
2%
.74
12%
.21
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-49
Demonstration Problem 18.5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-50
Demonstration Problem 18.5
OC Curven = 20; c = 2
0.00.10.20.30.40.50.60.70.80.91.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Fraction Defectives
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18-51
Demonstration Problem 18.5
n = 20c = 2
p P(Accept)0.00001 =BINOMDIST(B$2,B$1,A5,TRUE)0.01 =BINOMDIST(B$2,B$1,A6,TRUE)0.02 =BINOMDIST(B$2,B$1,A7,TRUE)0.03 =BINOMDIST(B$2,B$1,A8,TRUE)0.04 =BINOMDIST(B$2,B$1,A9,TRUE)0.05 =BINOMDIST(B$2,B$1,A10,TRUE)0.06 =BINOMDIST(B$2,B$1,A11,TRUE)0.07 =BINOMDIST(B$2,B$1,A12,TRUE)0.08 =BINOMDIST(B$2,B$1,A13,TRUE)0.09 =BINOMDIST(B$2,B$1,A14,TRUE)0.1 =BINOMDIST(B$2,B$1,A15,TRUE)0.11 =BINOMDIST(B$2,B$1,A16,TRUE)0.12 =BINOMDIST(B$2,B$1,A17,TRUE)0.13 =BINOMDIST(B$2,B$1,A18,TRUE)0.14 =BINOMDIST(B$2,B$1,A19,TRUE)0.15 =BINOMDIST(B$2,B$1,A20,TRUE)0.16 =BINOMDIST(B$2,B$1,A21,TRUE)0.17 =BINOMDIST(B$2,B$1,A22,TRUE)0.18 =BINOMDIST(B$2,B$1,A23,TRUE)0.19 =BINOMDIST(B$2,B$1,A24,TRUE)0.2 =BINOMDIST(B$2,B$1,A25,TRUE)