1. Chapter 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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Process Variation = + Total Process Variation Common Cause Variation
Assignable Cause 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.

7. Total Process Variation
Common
Cause Variation
Assignable Cause Variation = +
Variation is often due to differences in:
People
Machines
Materials
Methods
Measurement
Environment
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. Common Cause Variation
Total Process Variation
Common
Cause Variation
Assignable Cause 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.

9. Special Cause Variation
Total Process Variation
Common
Cause Variation
Assignable Cause 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.

10. 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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

12. Tools for Quality Improvement
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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. 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
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.

16. Individual measurements
Example: Subgroups
Sample measurements:
Subgroup measures
Subgroup number
Individual measurements
(subgroup size = 4)
Mean, x
Std. Dev., s 1 2 3 … 15 12 17 16 21 9 18 11 20
14.5
13.0
19.0
2.517
3.162
1.826
Average subgroup mean =
Average subgroup std. dev. = s
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. Factors for Control Charts
Selected control chart factors (Table 18.1) n c4 A3 B3 B4 2
.789
2.66
3.27 3
.886
1.95
2.57 4
.921
1.63
2.27 5
.940
1.43
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.

19. 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.

20. Control Charts and Control Limits
(continued)
The 3-standard-deviation control limits are estimated for an X-chart as follows:
Where the value of is given in Table 18.1 or in Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

23. X-Chart Example: Subgroup Data
Day
Subgroup Size
SubgroupMean
Subgroup Std. Dev. 1 2 3 4 5 6 7
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. X-Chart Control Limits Solution
A3 = 1.43 is from Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

25. X-Chart Control Chart Solution
Minutes
UCL = 8.889 8 _ _ 6
x = 5.813 4
LCL = 2.737 2 1 2 3 4 5 6 7
Day
Conclusion: Process mean is in statistical control
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

27. s-Chart Control Limits Solution
B4 and B3 are found in Appendix 13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. s-Chart Control Chart Solution
Minutes
UCL = 4.496 4 _ 2
s = 2.151
LCL = 0 1 2 3 4 5 6 7
Day
Conclusion: Variation is in control
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

29. Control Chart Basics UCL = Process Average + 3 Standard Deviations UCL
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.

30. Process Variability UCL = Process Average + 3 Standard Deviations UCL
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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

33. 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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

35. 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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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 (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.

40. 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
A satisfactory value is at least 1.33
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

41. 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.

42. Process Capability: Hotel Data
Day
Subgroup Size
Subgroup
Mean
Subgroup Std. Dev. 1 2 3 4 5 6 7
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

43. Process Capability: Hotel Example Solution
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.

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”
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

45. p-Chart Used with equal or unequal sample sizes (subgroups) over time
(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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

47. p-Chart Example
Subgroup number, i
Sample size
Number of successes
Sample Proportion, pi 1 2 3 …
150 15 12 17
.1000
.0800
.1133
Average sample proportions = p
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

48. Average of Sample Proportions
The average of sample proportions = p
If equal sample sizes:
where:
pi = sample proportion for subgroup i
K = number of subgroups of size n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

50. Computing Control Limits
(continued)
The upper and lower control limits for the p-chart are
Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

52. p Chart Example: Hotel Data
# Not Day # Rooms Ready Proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

53. p Chart Control Limits Solution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

54. p Chart Control Chart Solution
0.15
UCL = .1460 _
0.10
p = .0864
0.05
LCL = .0268
0.00 1 2 3 4 5 6 7
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

56. Mean and Standard Deviation for a c-Chart
The sample mean number of occurrences is
The standard deviation for a c-chart is
where:
ci = number of successes per item
K = number of items sampled
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

57. c-Chart Center and Control Limits
The center line for a c-chart is
The control limits for a c-chart are
The number of occurrences can never be negative, so if the calculated lower control limit is negative, set LCL = 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Flaws found
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

60. Constructing the c-Chart
The mean and standard deviation are:
The control limits are:
Note: LCL < 0 so set LCL = 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

61. The completed c-Chart UCL = 5.642 c = 1.714 LCL = 0 Sample number3 2 1
UCL = 5.642
c = 1.714
LCL = 0
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.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.