1624697595.pdf statistical quality control

Statistical quality control notes. Control charts for variables.R chart, X bar chart etc

1. Chapter 6 1Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

2. Chapter 6 2Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Learning Objectives

3. Chapter 6 3Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

4. Chapter 6 4Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

5. Chapter 6 5Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Subgroup Data with Unknown and 

6. Chapter 6 6Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

7. Chapter 6 7Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

8. Chapter 6 8Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

9. Chapter 6 9Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

10. Chapter 6 10Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Phase I Application of and RCharts
•Eqns 6.4 and 6.5 are trial control limits
–Determined from minitial samples
•Typically 20-25 subgroups of size nbetween 3 and 5
–Any out-of-control points should be examined for assignable causes
•If assignable causes are found, discard points from calculations and
revise the trial control limits
•Continue examination until all points plot in control
•Adopt resulting trial control limits for use
•If no assignable cause is found, there are two options
1.Eliminate point as if an assignable cause were found and revise limits
2.Retain point and consider limits appropriate for control
–If there are many out-of-control points they should be examined for
patternsthat may identify underlying process problemsx

11. Chapter 6 11Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Example 6.1 The Hard Bake Process

12. Chapter 6 12Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

13. Chapter 6 13Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

14. Chapter 6 14Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

15. Chapter 6 15Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

16. Chapter 6 16Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

17. Chapter 6 17Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Revision of Control Limits
and Center Lines
•Effective use of control charts requires periodic
review and revision of control limits and center lines
•Sometimes users replace the center line on the chart
with a target value
•When Rchart is out of control, out-of-control points
are often eliminated to recompute a revised value of
which is used to determine new limits and center line
on Rchart and new limits on chartx x R

18. Chapter 6 18Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Phase II Operation of Charts
•Use of control chart for monitoring future
production, once a set of reliable limits are
established, is called phase IIof control chart
usage (Figure 6.4)
•A run chart showing individuals observations
in each sample, called a tolerance chartor
tier diagram(Figure 6.5), may reveal patterns
or unusual observations in the data

19. Chapter 6 19Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

20. Chapter 6 20Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

21. Chapter 6 21Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

22. Chapter 6 22Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Control vs. Specification Limits
•Controllimits are derived
from natural process
variability, or the natural
tolerancelimits of a process
•Specificationlimits are
determined externally, for
example by customers or
designers
•There is no mathematical or
statistical relationship
between the control limits
and the specification limits

23. Chapter 6 23Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Rational Subgroups
•charts monitor between-sample variability
•Rcharts measure within-sample variability
•Standard deviation estimate of used to construct
control limits is calculated from within-sample
variability
•It is not correct to estimate usingx

24. Chapter 6 24Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Guidelines for Control Chart Design
•Control chart design requires specification of sample size,
control limit width, and sampling frequency
–Exact solution requires detailed information on statistical
characteristics as well as economic factors
–The problem of choosing sample size and sampling frequency is one of
allocating sampling effort
•For chart, choose as small a sample size is consistent with
magnitude of process shift one is trying to detect. For
moderate to large shifts, relatively small samples are effective.
For small shifts, larger samples are needed.
•For small samples, Rchart is relatively insensitive to changes
in process standard deviation. For larger samples (n> 10 or
12), sor s
2
charts are better choices.x

25. Chapter 6 25Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

26. Chapter 6 26Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

27. Chapter 6 27Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
6.2.4 Interpretation of Control Charts

28. Chapter 6 28Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

29. Chapter 6 29Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
•An assumption in performance properties is that the
underlying distribution of quality characteristic is normal
–If underlying distribution is not normal, sampling distributions can be
derived and exact probability limits obtained
•Burr (1967) notes the usual normal theory control limits are
very robust to normality assumption
•Schilling and Nelson (1976) indicate that in most cases,
samples of size 4 or 5 are sufficient to ensure reasonable
robustness to normality assumption for chart
•Sampling distribution of Ris notsymmetric, thus symmetric
3-sigma limits are an approximation and -risk is not 0.0027.
Rchart is more sensitive to departures from normality than
chart.
•Assumptions of normality and independence are not a primary
concern in phase Ix x
6.2.5 The Effect of Non-normality

30. Chapter 6 30Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
6.2.6 The Operating Characteristic Function

31. Chapter 6 31Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
If the shift is 1.0σand the sample
size is n= 5, then β= 0.75.

32. Chapter 6 32Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

33. Chapter 6 33Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

34. Chapter 6 34Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

35. Chapter 6 35Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

36. Chapter 6 36Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

37. Chapter 6 37Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

38. Chapter 6 38Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Development of the control limits:
Thius produces the control limits in equation (6.27)

39. Chapter 6 39Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
This produces the control limits in equation (6.28)

40. Chapter 6 40Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

41. Chapter 6 41Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

42. Chapter 6 42Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

43. Chapter 6 43Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

44. Chapter 6 44Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

45. Chapter 6 45Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

46. Chapter 6 46Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

47. Chapter 6 47Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

48. Chapter 6 48Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

49. Chapter 6 49Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

50. Chapter 6 50Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

51. Chapter 6 51Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

52. Chapter 6 52Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

53. Chapter 6 53Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

54. Chapter 6 54Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

55. Chapter 6 55Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

56. Chapter 6 56Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Average Run Lengths
•Crowder (1987b) showed that
ARL
0of combined individuals
and moving-range chart with
conventional 3-sigma limits is
generally much less than ARL
0
(= 370) of standard Shewhart
control chart
•Ability of individuals chart to
detect small shifts is very poor
–Rather than narrowing the 3-
sigma limits, correct approach to
detecting small shifts is a
cumulative-sum or exponentially
weighted moving-average control
chart (Chapter 9)

57. Chapter 6 57Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Normality
•Borror, Montgomery, and Runger (1999) report that the in-control ARL is
dramatically affected by nonnormal data
•One approach for nonnormal data is to determine control limits for individuals
control chart based on percentiles of correct underlying distribution
–Requires at least 100 and preferably 200 observations
–Transformations can also be useful

58. Chapter 6 58Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

59. Chapter 6 59Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

60. Chapter 6 60Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

61. Chapter 6 61Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

62. Chapter 6 62Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

63. Chapter 6 63Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

64. Chapter 6 64Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.

65. Chapter 6 65Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2012John Wiley & Sons, Inc.
Learning Objectives