Statistics for Business and Economics

1
Statistics for Business and Economics

• Chapter 12
• Methods for Quality Improvement

2
Learning Objectives

• Define Quality
• Describe Types of Variation
• Explain Control Charts
• xChart
• RChart
• pChart

3
What is Quality?

• Performance
• Primary operating characteristics of the product
• Features
• "bells and whistles"
• Reliability
• Probability product will function for a specified
  amount of time

4
What is Quality?

• Conformance
• Extent to which product meets preestablished
  standards
• Durability
• The life of the product
• Serviceability
• Ease and speed of repair

5
What is Quality?

• Aesthetics
• Way product looks, feels, etc.
• Other Perceptions
• e.g. Company reputation

6
Process
InputsInformationMethodsEnergyMaterialsMachin
esPeople
OutputsFinished Products
Variability is present in the output of all
processes
7
System
Supplier
Customer
8
Sources of Variation

• People
• Machines
• Materials
• Methods
• Measurement
• Environment

9
Types of Variation

• Common causes
• Methods, materials, people, environment
• Special causes
• Single worker, bad batch of material
• Only detectable when process is incontrol
  (stable)

10
Time Series Plot (Run Chart)

• Graphically shows trends and changes in the data
  over time
• Time recorded on the horizontal axis
• Measurements recorded on the vertical axis
• Points connected by straight lines

11
Time Series Pattern Random Behavior
centerline
Measurement
Order of production
12
Time Series Pattern Shift
centerline
Measurement
Order of production
13
Time Series Pattern Increased Variance
centerline
Measurement
Order of production
14
Control Chart Uses

• Monitor process variation
• Differentiate between variation due to common
  causes v. special causes
• Evaluate past performance
• Monitor current performance

15
Sample Control Chart
Upper control limit
centerline
Measurement
Lower control limit
Order of production
16
Control Limits

• 3sigma limits
• Upper control limit µ 3s
• Lower control limit µ 3s

.00135
.00135
Order of production
17
4 Possible Outcomes

• H0 Process is in control
• Ha Process is out of control

Reality
Conclude process is in control
H0 True
Ha True
Correct decision
Type II Error
H0 True
Conclusion
Correct decision
Type I Error
Ha True
Conclude process is out of control
18
Types of Control Charts
19
The xChart
20
Types of Control Charts
21
xChart

• Monitors changes in the mean of samples
• Horizontal axis Sample number
• Vertical axis Mean of sample
• Control limits based on sampling distribution of
  x
• Standard deviation of x

22
Sample xChart
Upper control limit
centerline
x
Lower control limit
Sample Number
23
Determining the Centerline
k number of samples of size n (usually between
2 and 10)
xi sample mean of the ith sample
24
Estimating s

• Determine the Range of each sample Range
  Maximum Minimum
• Determine the Average Range of the k samples
• Divide R by the constant d2 (based on sample
  size)

25
Determining the Control Limits
where
26
xChart Summary

• Collect at least 20 samples of size 2 to 10
• Calculate mean and range of each sample
• Determine the centerline and control limits

where
27
xChart Example
Samples from a machine filling 12oz soda cans
28
xChart Centerline Solution
29
xChart Control Limits Solution
30
xChart Control Limits Solution
31
xChart Solution
UCL 12.61
LCL 11.38
32
Interpreting Control Charts

• Six zones
• Each zone is one standard deviation wide

UCL
Zone A
Zone B
Zone C
centerline
Zone C
Zone B
Zone A
LCL
Order of production
33
Zone Boundaries

• 3sigma control limit zone boundaries

34
Zone Boundaries Example
Samples from a machine filling 12oz soda cans
35
Zone AB Boundaries Solution

• Recall

Upper AB
Lower AB
36
Zone BC Boundaries Solution

• Recall

Upper BC
Lower BC
37
xChart Solution
UCL 12.61
A
12.4
B
12.2
C
C
11.8
B
11.6
A
LCL 11.38
38
Pattern Analysis Rules

• Rule 1 One point beyond Zone A
• Either lower or upper half of control chart

39
Pattern Analysis Rules

• Rule 2 Nine points in a row in Zone C or beyond
• Either lower or upper half of control chart

40
Pattern Analysis Rules

• Rule 3 Six points in a row steadily increasing
  or decreasing

41
Pattern Analysis Rules

• Rule 4 Fourteen points in a row alternating up
  and down

42
Pattern Analysis Rules

• Rule 5 Two out of three points in Zone A or
  beyond
• Either lower or upper half of control chart

43
Pattern Analysis Rules

• Rule 6 Four out of five points in Zone B or
  beyond
• Either lower or upper half of control chart

44
Interpreting an xChart

• Process is considered out of control if any of
  the pattern analysis rules are detected
• Process is considered in control if none of the
  pattern analysis rules are detected

45
Interpreting xChart Example
What does the chart suggest about the stability
of the process?
UCL 12.61
A
12.4
B
12.2
C
C
11.8
B
11.6
A
LCL 11.38
46
Interpreting xChart Solution
Since none of the six pattern analysis rules are
observed, the process is considered in control
47
Interpreting xChart Thinking Challenge
Ten additional samples of size 5 are taken. What
does the chart suggest about the stability of the
process?
UCL 12.61
A
12.4
B
12.2
C
C
11.8
B
11.6
A
LCL 11.38
48
Interpreting xChart Solution
Rule 5 and Rule 6 are violated. Process is out of
control
UCL 12.61
A
12.4
B
12.2
C
C
11.8
B
11.6
A
LCL 11.38
49
RChart
50
Types of Control Charts
Type of Data
QuantitativeData
QualitativeData
xchart
51
RChart

• Monitors changes in process variation
• Horizontal axis Sample number
• Vertical axis Sample ranges
• Control limits based on sampling distribution of
  R
• Mean of sampling distribution of R µR
• Standard deviation of sampling distribution of R
  sR

52
Estimating µR and sR
Estimate of µR
k number of samples of size n 2
Ri sample range of the ith sample
53
Determining the Control Limits
Note If n 6, the LCL will be negative. Since
the range cant be negative the LCL is
meaningless.
54
RChart Summary

• Collect at least 20 samples of size n 2
• Calculate the range of each sample
• Determine the centerline and control limits

where
55
Zone Boundaries
56
Interpreting an RChart

• Process is considered out of control if any of
  the pattern analysis rules 1 4 are detected
• One point beyond Zone A
• Nine points in a row in Zone C or beyond
• Six points in a row steadily increasing or
  decreasing
• Fourteen points in a row alternating up and down
• Process is considered in control if none of the
  pattern analysis rules are detected

57
RChart Example
Samples from a machine filling 12oz soda cans
58
RChart Solution
Calculate the mean of the ranges
59
RChart Solution
Calculate the control limits.
n 5 D4 2.114 D3 0 (LCL will be
zero)
60
RChart Solution
Determine the AB zone boundaries
Upper AB Boundary
Lower AB Boundary
61
RChart Solution
Determine the BC zone boundaries
Upper BC Boundary
Lower BC Boundary
62
RChart Solution
UCL 2.3
A
1.9
B
1.5
C
C
.7
B
.3
A
LCL 0
The variation of the process is in control
63
pChart
64
Types of Control Charts
65
pChart

• Used for qualitative data
• Monitors variation in the process proportion
• Horizontal axis Sample number
• Vertical axis Sample proportion
• Control limits based on sampling distribution of
  p
• Mean of sampling distribution of p µp
• Standard deviation of sampling distribution of p
  sp

66
Estimating µp and sp


67
Determining the Control Limits
Note If the LCL is negative do not plot it on
the control chart.
68
pChart Summary

• Collect at least 20 samples of size
• Calculate the proportion of defective units in
  each sample
• Determine the centerline and control limits

p0 is an estimate of p
69
Zone Boundaries
70
Interpreting a pChart

• Process is considered out of control if any of
  the pattern analysis rules 1 4 are detected
• One point beyond Zone A
• Nine points in a row in Zone C or beyond
• Six points in a row steadily increasing or
  decreasing
• Fourteen points in a row alternating up and down
• Process is considered in control if none of the
  pattern analysis rules are detected

71
pChart Example
A manufacturer of pencils knows about 4 of
pencils produced fail to meet specifications. How
many pencils should be sampled for monitoring the
process proportion?
?
?
?
Solution
Samples of size 216 or more should be selected.
72
pChart Example
The pencil manufacturer has decided to select
samples of size n 225. The table shows the
results for the past 20 samples. Construct a
pchart.
?
?
?
73
pChart Solution
Calculate the centerline
74
pChart Solution
Calculate the control limits
Since LCL is negative, do not plot it on the
control chart
75
pChart Solution
Determine the AB zone boundaries
Upper AB Boundary .06407
Lower AB Boundary .01281
76
pChart Solution
Determine the BC zone boundaries
Upper BC Boundary .05126
Lower BC Boundary .02562
77
pChart Solution
UCL .07689
A
.06407
B
.05126
C
C
.02562
B
.01281
A
The process is in control
78
Conclusion

• Defined Quality
• Described Types of Variation
• Explained Control Charts
• xChart
• RChart
• pChart