1. Statistics for Business and Economics
Chapter 12
Methods for Quality Improvement

2. Learning Objectives Define Quality Describe Types of Variation
Explain Control Charts
x–Chart
R–Chart
p–Chart
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3. What is Quality? Performance Features Reliability
Primary operating characteristics of the product
Features
"bells and whistles"
Reliability
Probability product will function for a specified amount of time
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4. What is Quality? Conformance Durability Serviceability
Extent to which product meets preestablished standards
Durability
The life of the product
Serviceability
Ease and speed of repair
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5. What is Quality? Aesthetics Other Perceptions
Way product looks, feels, etc.
Other Perceptions
e.g. Company reputation
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6. Process Process Operations
Inputs Information Methods Energy Materials Machines People
Outputs Finished Products
Variability is present in the output of all processes

7. System
Processes
System
Supplier
Customer
Feedback

8. Sources of Variation People Machines Materials Methods Measurement
Environment
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9. Types of Variation Common causes Special causes
Methods, materials, people, environment
Special causes
Single worker, bad batch of material
Only detectable when process is in–control (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 3–sigma limits Upper control limit: μ + 3σ
Lower control limit: μ – 3σ
.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
Use 3–sigma limits because of small P(Type I Error)="False Alarm"
Correct decision
Type II Error
H0 True
Conclusion
Type I Error
Correct decision
Ha True
Conclude process is out of control

18. Types of Control Charts
Type of Data
Quantitative Data
Qualitative Data
R–chart
x–chart
p–chart 3

19. The x–Chart

20. Types of Control Charts
Type of Data
Quantitative Data
Qualitative Data
R–chart
x–chart
p–chart 3

21. x–Chart 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 x–Chart 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 σ
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. x–Chart 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. x–Chart Example
Samples from a machine filling 12oz soda cans

28. x–Chart Centerline Solution

29. x–Chart Control Limits Solution

30. x–Chart Control Limits Solution

31. x–Chart Solution
UCL = 12.61
LCL = 11.38

32. Interpreting Control Charts
Six zones
Each zone is one standard deviation wide
Upper A–B boundary
UCL
Zone A
Upper B–C boundary
Zone B
Zone C
centerline
Lower B–C boundary
Zone C
Zone B
Lower A–B boundary
Zone A
LCL
Order of production

33. Zone Boundaries 3–sigma control limit zone boundaries:
Upper A–B Boundary:
Lower A–B Boundary:
Upper B–C Boundary:
Lower B–C Boundary:

34. Zone Boundaries Example
Samples from a machine filling 12oz soda cans

35. Zone A–B Boundaries Solution
Recall
Upper A–B:
Lower A–B:

36. Zone B–C Boundaries Solution
Recall
Upper B–C:
Lower B–C:

37. x–Chart Solution UCL = 12.61 12.4 12.2 11.8 11.6 LCL = 11.38 A B C C B

38. Using JMP for Control Charts
Graph >> Control Chart >> Xbar
But for JMP…Data must be in vertical format…so we need to first stack the weight columns into a single column

39. Using JMP for Control Charts
Tables >> Stack (now specify the columns Weight1 through Weight6 in the Stack Columns box)
Result

40. Using JMP for Control Charts
Now do chart (Graph >> Control Chart >> Xbar)
Result

41. Pattern Analysis Rules
Rule 1: One point beyond Zone A
Either lower or upper half of control chart
centerline
LCL
UCL A B C

42. Pattern Analysis Rules
Rule 2: Nine points in a row in Zone C or beyond
Either lower or upper half of control chart
centerline
LCL
UCL A B C

43. Pattern Analysis Rules
Rule 3: Six points in a row steadily increasing or decreasing
centerline
LCL
UCL A B C

44. Pattern Analysis Rules
Rule 4: Fourteen points in a row alternating up and down
centerline
LCL
UCL A B C

45. Pattern Analysis Rules
Rule 5: Two out of three points in Zone A or beyond
Either lower or upper half of control chart
centerline
LCL
UCL A B C

46. Pattern Analysis Rules
Rule 6: Four out of five points in Zone B or beyond
Either lower or upper half of control chart
centerline
LCL
UCL A B C

47. Interpreting an x–Chart
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

48. Interpreting x–Chart 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

49. Interpreting x–Chart Solution
Since none of the six pattern analysis rules are observed, the process is considered in control

50. Interpreting x–Chart 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

51. Interpreting x–Chart 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

52. Using JMP for Pattern Tests
JMP gives 8 tests---these six and two more.
To get them, use the red triangle pull down menu and choose “Tests”

53. R–Chart

54. Types of Control Charts
Type of Data
Quantitative Data
Qualitative Data
x–chart
R–chart
p–chart 3

55. R–Chart 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: σR

56. Estimating μR and σR Estimate of μR:
k = number of samples of size n ≥ 2
Ri = sample range of the ith sample
Estimate of σR:

57. Determining the Control Limits
Note: If n ≤ 6, the LCL will be negative. Since the range can’t be negative the LCL is meaningless.

58. R–Chart Summary Collect at least 20 samples of size n ≥ 2
Calculate the range of each sample
Determine the centerline and control limits
where

59. Zone Boundaries Upper A–B Boundary: Lower A–B Boundary:
Upper B–C Boundary:
Lower B–C Boundary:

60. Interpreting an R–Chart
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

61. R–Chart Example
Samples from a machine filling 12oz soda cans

62. R–Chart Solution
Calculate the mean of the ranges:

63. R–Chart Solution Calculate the control limits.
n = D4 = D3 = 0 (LCL will be zero)

64. R–Chart Solution Determine the A–B zone boundaries Upper A–B Boundary:
Lower A–B Boundary:

65. R–Chart Solution Determine the B–C zone boundaries Upper B–C Boundary:
Lower B–C Boundary:

66. R–Chart Solution The variation of the process is in control UCL = 2.3
1.9 B
1.5 C C .7 B .3 A
LCL = 0
The variation of the process is in control

67. p–Chart

68. Types of Control Charts
Type of Data
Quantitative Data
Qualitative Data
R–chart
x–chart
p–chart 3

69. p–Chart 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: σp ^ ^ ^ ^ ^

70. Estimating μp and σp ^ ^ Estimate of μp:
Total number of defective units in all k samples
Total number of units sampled
p =
Estimate of σp: ^

71. Determining the Control Limits
Note: If the LCL is negative do not plot it on the control chart.

72. p–Chart 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
Number of defective units in the sample
Number of units in the sample
p = ^

73. Zone Boundaries Upper A–B Boundary: Lower A–B Boundary:
Upper B–C Boundary:
Lower B–C Boundary:

74. Interpreting a p–Chart
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

75. p–Chart 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.

76. p–Chart 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 p–chart.   

77. p–Chart Solution
Calculate the centerline:

78. p–Chart Solution Calculate the control limits:
*Since LCL is negative, do not plot it on the control chart

79. p–Chart Solution Determine the A–B zone boundaries
Upper A–B Boundary:
Lower A–B Boundary:

80. p–Chart Solution Determine the B–C zone boundaries
Upper B–C Boundary:
Lower B–C Boundary:

81. p–Chart Solution The process is in control UCL = .07689 .06407 .05126B
.05126 C C
.02562 B
.01281 A
The process is in control

82. Graph >> Control Charts >> P
JMP p–Chart Solution
Graph >> Control Charts >> P

83. Conclusion Defined Quality Described Types of Variation
Explained Control Charts
x–Chart
R–Chart
p–Chart
As a result of this class, you will be able to ...