1. Chapter 4 Control Charts for Measurements with Subgrouping (for One Variable)

2. Presumptions Subgroups (samples) of data are formed.
Measurements are made and the values are obtained with sufficient speed.

3. 4.1 Basic Control Chart Principles
Stage 1: Control charts can be used to determine if a process has been in a state of statistical control by examining past data. (retrospective data analysis)
Stage 2: Recent data can be used to determine control limits that would apply to future data obtained from a process.

4. Basic Control Chart Principles
Control charts alone cannot produce statistical control
Control charts can indicate whether or not statistical control is being maintained and provide users with other signals from the data
To avoid unnecessary and undesired process adjustments
Control charts can also be used to study process capability (Chapter 7)

5. Basic Control Chart Principles
Best results will generally be obtained when control charts are applied primarily to process variables than to product variables.
In general, it is desirable to monitor all process variables that affect important product variables.
Control charts are essentially plots of data over time.

6. Figure 4.1

7. Basic Control Chart Principles
Procedures for setting up new control charts
Obtain at least 20 subgroups or at least 100 individual observations (from past or current data)
Calculate the trial control limits
Data points outside the trial limits should be investigated. Those points can only be removed if the assignable causes can be detected and removed
Repeat steps 2 and 3 until no further action can be taken

8. Basic Control Chart Principles
After a process has been brought into a state of statistical control, a process capability study can be initiated to determine the capability of the process in regard to meeting the specifications.
Process performance indices: using long term sigma (when a process is not in a state of statistical control)

9. 4.2 Real-time Control Charting vs Analysis of Past Data
When a set of points is plotted all at once, the probability of observing at least one point that is outside the control limit will be much greater than
When points are plotted individually,
3-sigma limits are used,
A normal distribution is assumed, and
Parameter of the appropriate distribution are assumed to be known

10. Table 4.1 Probabilities of Points Plotting Outside Control Limits
Actual Prob. 1
0.0027 2
0.0054 5
0.0135
0.0134 10
0.0270
0.0267 15
0.0405
0.0397 20
0.0540
0.0526 25
0.0675
0.0654 50
0.1350
0.1264
100
0.2700
0.2369
350
0.9450
0.6118

11. Real-time Control Charting vs Analysis of Past Data
The calculation of probabilities can only be done if the parameters are assumed to be known.
In general, the possible use of k-sigma limits in Stage 1 needs to be addressed.
Costs of shutting down the process
Costs of making units outside specifications
Costs of false looking for assignable causes
Medical practitioners often tend to favor 2-sigma limits. (assignable causes can be detected as quickly as possible)

12. 4.3 Control Charts: When to Use, Where to Use, How many to Use
Not at every work station
Nature of the product often preclude measurements
No need for control charts in a process that is highly unlikely the process ever go out of control
Control charts should be used where trouble is likely to occur
They should be used where the potential for cost reduction is substantial
Number of charts
Computer vs manual charting

13. 4.4 Benefits from the Use of Control Charts
Good record keeping
Control charts work as an aid in identifying special causes of variation

14. 4.5 Rational Subgroups
Important Requirement: data chosen for the subgroups come from the same population (same operator, shift, machine, etc.)
If data are mixed, the control limits will correspond to a mixture (aggregated) distribution

15. 4.6 Basic Statistical Aspects of Control Charts

16. Basic Statistical Aspects of Control Charts
When X is highly asymmetric, data can usually be transformed (log, square root, power, reciprocal, Cox-cox, etc.) to be approximately normal.

17. 4.7 Illustrative Example
Important Requirement: data chosen for the subgroups come from the same population (same operator, shift, machine, etc.)
If data are mixed, the control limits will correspond to a mixture (aggregated) distribution

18. Table 4.2 Data in Subgroups Obtained at Regular IntervalsX1 X2 X3 X4 1 72 84 79 49 2 56 87 33 42 3 55 73 22 60 4 44 80 54 74 5 97 26 48 58 6 83 89 91 62 7 47 66 53 8 88 50 69 9 57 41 46 10 13 30 32 11 39 52 12 27 63 34 78 14 71 82 15 70 16 67 94 17 18 51 76 19 61 59 20
X-bar R S
71.00 35
15.47
54.50 54
23.64
52.50 51
21.70
63.00 36
16.85
57.25 71
29.68
81.25 29
13.28
56.00 19
8.04
72.75 38
17.23
47.75 16
6.70
21.25 22
11.35
41.25 26
11.53
42.50
15.76
69.00
16.87
67.75 27
67.00 13
6.06
75.00
12.73
59.75 23
9.98
57.75
12.42
68.00 21
9.83
63.50 17
7.33

19. R vs. S Charts
Either R or S charts could be used in controlling the process variability.
S-chart is preferable since it uses all the observations in each subgroup.
Other statistical methods in quality improvement are generally based on S (or S2)

20. Estimating of Population Parameters by Sample Statistics
Population Statistics:  , usually unknown
Using Sample Statistics to estimate population statistics:
Point estimates

21. 4.7.1 R-Chart

22. Figure 4.2 R-Chart

23. 4.7.2 R-Chart with Probability Limits

24. 4.7.3 S-Chart

25. S-Chart

26. 4.7.4 S-Chart with Probability Limits

27. Table 4.3 The .001 and .999 Percentage Points of 2 Distribution 3
0.0020 4
0.0243 5
0.0908 6
0.2102 7
0.3811 8
0.5985 9
0.8571 10
1.1519 11
1.4787 12
1.8339 13
2.2142 14
2.6172 15
3.0407

28. S-Chart with Probability Limits

29. 4.7.5 S2-Chart

32. Questions to ask:
What is the likelihood to have one of 20 sample data fall outside the control limits?
What is the probability for an subgroup average as small as 21.25?

34. 4.7.7 Recomputing Control Limits

35. 4.7.8 Applying Control Limits to Future Production