1. Yandell – Econ 216 Chapter 17 Statistical Applications in Quality Management Chap 17-1

2. Yandell – Econ 216 Chapter Goals After completing this chapter, you should be able to:  Use the seven basic tools of quality  Construct and interpret x-bar and R-charts  Construct and interpret p-charts  Construct and interpret c-charts Chap 17-2

3. Yandell – Econ 216 Chapter Overview Quality Management and Tools for Improvement Deming’s 14 Points Juran’s 10 Steps to Quality Improvement The Basic 7 Tools Philosophy of Quality Tools for Quality Improvement Control Charts X-bar/R-charts p-charts c-charts Chap 17-3

4. Yandell – Econ 216 Themes of Quality Management Primary focus is on process improvement Most variations in process are due to systems Teamwork is integral to quality management Customer satisfaction is a primary goal Organization transformation is necessary It is important to remove fear Higher quality costs less Chap 17-4

5. Yandell – Econ 216 1. Create a constancy of purpose toward improvement become more competitive, stay in business, and provide jobs 2. Adopt the new philosophy Better to improve now than to react to problems later 3. Stop depending on inspection to achieve quality -- build in quality from the start Inspection to find defects at the end of production is too late 4. Stop awarding contracts on the basis of low bids Better to build long-run purchaser/supplier relationships Deming’s 14 Points Chap 17-5

6. Yandell – Econ 216 5. Improve the system continuously to improve quality and thus constantly reduce costs 6. Institute training on the job Workers and managers must know the difference between common cause and special cause variation 7. Institute leadership Know the difference between leadership and supervision 8. Drive out fear so that everyone may work effectively. 9. Break down barriers between departments so that people can work as a team. (continued) Deming’s 14 Points Chap 17-6

7. Yandell – Econ 216 10. Eliminate slogans and targets for the workforce They can create adversarial relationships 11. Eliminate quotas and management by objectives 12. Remove barriers to pride of workmanship 13. Institute a vigorous program of education and self-improvement 14. Make the transformation everyone’s job (continued) Deming’s 14 Points Chap 17-7

8. Yandell – Econ 216 Juran’s 10 Steps to Quality Improvement 1. Build awareness of both the need for improvement and the opportunity for improvement 2. Set goals for improvement 3. Organize to meet the goals that have been set 4. Provide training 5. Implement projects aimed at solving problems Chap 17-8

9. Yandell – Econ 216 Juran’s 10 Steps to Quality Improvement 6. Report progress 7. Give recognition 8. Communicate the results 9. Keep score 10. Maintain momentum by building improvement into the company’s regular systems (continued) Chap 17-9

10. Yandell – Econ 216 The Deming Cycle The Deming Cycle The key is a continuous cycle of improvement Act Plan Do Study Chap 17-10

11. Yandell – Econ 216 The Basic 7 Tools 1.Process Flowcharts 2.Brainstorming 3.Fishbone Diagram 4.Histogram 5.Trend Charts 6.Scatter Plots 7.Statistical Process Control Charts Chap 17-11

12. Yandell – Econ 216 The Basic 7 Tools 1.Process Flowcharts 2.Brainstorming 3.Fishbone Diagram 4.Histogram 5.Trend Charts 6.Scatter Plots 7.Statistical Process Control Charts Map out the process to better visualize and understand opportunities for improvement (continued) Chap 17-12

13. Yandell – Econ 216 The Basic 7 Tools 1.Process Flowcharts 2.Brainstorming 3.Fishbone Diagram 4.Histogram 5.Trend Charts 6.Scatter Plots 7.Statistical Process Control Charts Cause 4 Cause 3 Cause 2Cause 1 Problem Fishbone (cause-and-effect) diagram: Sub-causes Show patterns of variation (continued) Chap 17-13

14. Yandell – Econ 216 The Basic 7 Tools 1.Process Flowcharts 2.Brainstorming 3.Fishbone Diagram 4.Histogram 5.Trend Charts 6.Scatter Plots 7.Statistical Process Control Charts time y x y Identify trend Examine relationships (continued) Chap 17-14

15. Yandell – Econ 216 The Basic 7 Tools 1.Process Flowcharts 2.Brainstorming 3.Fishbone Diagram 4.Histogram 5.Trend Charts 6.Scatter Plots 7.Statistical Process Control Charts X Examine the performance of a process over time time (continued) Chap 17-15

16. Yandell – Econ 216 Introduction to Control Charts Control Charts are used to monitor variation in a measured value from a process Exhibits trend Can make correction before process is out of control A process is a repeatable series of steps leading to a specific goal Inherent variation refers to process variation that exists naturally. This variation can be reduced but not eliminated Chap 17-16

17. Yandell – Econ 216 Process Variation Total Process Variation Common Cause Variation Special 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 Chap 17-17

18. Yandell – Econ 216 Sources of Variation Total Process Variation Common Cause Variation Special Cause Variation =+ People Machines Materials Methods Measurement Environment Variation is often due to differences in: Chap 17-18

19. Yandell – Econ 216 Common Cause Variation Total Process Variation Common Cause Variation Special Cause Variation =+ Common cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environment Chap 17-19

20. Yandell – Econ 216 Special Cause Variation Total Process Variation Common Cause Variation Special Cause Variation =+ Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered inherent to the process Chap 17-20

21. Yandell – Econ 216 Statistical Process Control Charts Show when changes in data are due to: Special or assignable causes Fluctuations not inherent to a process Represents problems to be corrected Data outside control limits or trend Common causes or chance Inherent random variations Consist of numerous small causes of random variability Chap 17-21

22. Yandell – Econ 216 Process Average Control Chart Basics UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL LCL +3σ - 3σ- 3σ Common Cause Variation: range of expected variability Special Cause Variation: Range of unexpected variability time Chap 17-22

23. Yandell – Econ 216 Process Average Process Variability UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations UCL LCL ±3σ → 99.7% of process values should be in this range time Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present Chap 17-23

24. Yandell – Econ 216 Statistical Process Control Charts X-bar charts and R-charts c-charts Used for measured numeric data Used for proportions (attribute data) Used for number of attributes per sampling unit p-charts Chap 17-24

25. Yandell – Econ 216 x-bar chart and R-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 Chap 17-25

26. Yandell – Econ 216 Steps to create an x-chart and an R-chart Calculate subgroup means and ranges Compute the average of the subgroup means and the average range value Prepare graphs of the subgroup means and ranges as a line chart Chap 17-26

27. Yandell – Econ 216 Steps to create an x-chart and an R-chart Compute the upper and lower control limits for the x-bar chart Compute the upper and lower control limits for the R-chart Use lines to show the control limits on the x-bar and R-charts (continued) Chap 17-27

28. Yandell – Econ 216 Example: x-chart Process measurements: Subgroup measures Subgroup number Individual measurements Mean, x Range, R 123…123… 15 12 17 … 17 16 21 … 15 9 18 … 11 15 20 … 14.5 13.0 19.0 … 674…674… Average subgroup mean = x Average subgroup range = R Chap 17-28

29. Yandell – Econ 216 Average of Subgroup Means and Ranges Average of subgroup means: where: x i = i th subgroup average k = number of subgroups Average of subgroup ranges: where: R i = i th subgroup range k = number of subgroups Chap 17-29

30. Yandell – Econ 216 Computing Control Limits The upper and lower control limits for an x-chart are generally defined as or UCL = Process Average + 3 Standard Deviations LCL = Process Average – 3 Standard Deviations Chap 17-30

31. Yandell – Econ 216 Computing Control Limits Since control charts were developed before it was easy to calculate σ, the interval was formed using R instead The value A 2 R is used to estimate 3σ, where A 2 is from Appendix Table E.11 The upper and lower control limits are (continued) where A 2 = Shewhart factor for subgroup size n from appendix Q Chap 17-31

32. Yandell – Econ 216 Example: R-chart The upper and lower control limits for an R-chart are where: D 4 and D 3 are taken from the Shewhart table (Appendix Table E.11) for subgroup size = n Chap 17-32

33. Yandell – Econ 216 x-chart and R-chart UCL LCL time UCL LCL time R-chart x-chart Chap 17-33

34. Yandell – Econ 216 Using Control Charts Control Charts are used to check for process control H 0 : The process is in control i.e., variation is only due to common causes H A : The process is out of control i.e., special cause variation exists If the process is found to be out of control, steps should be taken to find and eliminate the special causes of variation Chap 17-34

35. Yandell – Econ 216 Process In Control Process in control: points are randomly distributed around the center line and all points are within the control limits UCL LCL time Chap 17-35

36. Yandell – Econ 216 Process Not in Control Out of control conditions: One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line Chap 17-36

37. Yandell – Econ 216 Process Not in Control One or more points outside control limits UCL LCL Nine or more points in a row on one side of the center line UCL LCL Six or more points moving in the same direction UCL LCL 14 or more points alternating above and below the center line UCL LCL Chap 17-37

38. Yandell – Econ 216 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 Chap 17-38

39. Yandell – Econ 216 p-Chart Control chart for proportions Is an attribute chart Shows proportion of 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” Chap 17-39

40. Yandell – Econ 216 p-Chart 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 np > 5 and n(1-p) > 5 (continued) Chap 17-40

41. Yandell – Econ 216 Creating a p-Chart Calculate subgroup proportions Compute the average of the subgroup proportions Prepare graphs of the subgroup proportions as a line chart Compute the upper and lower control limits Use lines to show the control limits on the p-chart Chap 17-41

42. Yandell – Econ 216 p-Chart Example Subgroup number Sample size Number of successes Proportion, p 123…123… 150 15 12 17 … 10.00 8.00 11.33 … Average subgroup proportion = p Chap 17-42

43. Yandell – Econ 216 Average of Subgroup Proportions The average of subgroup proportions = p where: p i = sample proportion for subgroup i k = number of subgroups of size n where: n i = number of items in sample i  n i = total number of items sampled in k samples If equal sample sizes:If unequal sample sizes: Chap 17-43

44. Yandell – Econ 216 Computing Control Limits The upper and lower control limits for an p-chart are or UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion – 3 Standard Deviations Chap 17-44

45. Yandell – Econ 216 Standard Deviation of Subgroup Proportions The estimate of the standard deviation for the subgroup proportions is If equal sample sizes:If unequal sample sizes: where: = mean subgroup proportion n = common sample size Generally, is computed separately for each different sample size Chap 17-45

46. Yandell – Econ 216 Computing Control Limits The upper and lower control limits for the p-chart are (continued) If sample sizes are equal, this becomes Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0 Chap 17-46

47. Yandell – Econ 216 p-Chart Examples For equal sample sizes For unequal sample sizes UCL LCL UCL LCL pp is constant since n is the same for all subgroups varies for each subgroup since n i varies Chap 17-47

48. Yandell – Econ 216 c-Chart Control chart for number of nonconformities (occurrences) per sampling unit (an area of opportunity) 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 Chap 17-48

49. Yandell – Econ 216 Mean and Standard Deviation for a c-Chart The mean for a c-chart is The standard deviation for a c-chart is where: x i = number of successes per sampling unit k = number of sampling units Chap 17-49

50. Yandell – Econ 216 c-Chart Control Limits The control limits for a c-chart are Chap 17-50

51. Yandell – Econ 216 Process Control Determine process control for p-chars and c-charts using the same rules as for x-bar and R-charts Out of control conditions: One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line Chap 17-51

52. Yandell – Econ 216 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 number1234567 Flaws found2130510 Chap 17-52

53. Yandell – Econ 216 Constructing the c-Chart The mean and standard deviation are: The control limits are: Note: LCL < 0 so set LCL = 0 Chap 17-53

54. Yandell – Econ 216 The completed c-Chart 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 UCL = 5.642 LCL = 0 Sample number 1 2 3 4 5 6 7 c = 1.714 65432106543210 Chap 17-54

55. Yandell – Econ 216 Chapter Summary Reviewed the philosophy of quality management Demings 14 points Juran’s 10 steps Described the seven basic tools of quality Discussed the theory of control charts Common cause variation vs. special cause variation  Constructed and interpreted x-bar and R-charts  Constructed and interpreted p-charts  Constructed and interpreted c-charts Chap 17-55