Transcript Chapter 1 Making Economic Decisions

Chapter 4
Control Charts for
Measurements with Subgrouping
(for One Variable)
Presumptions
• Subgroups (samples) of data are formed.
• Measurements are made and the values are
obtained with sufficient speed.
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.
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)
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.
Figure 4.1
Basic Control Chart Principles
Procedures for setting up new control charts
1.Obtain at least 20 subgroups or at least 100 individual
observations (from past or current data)
2.Calculate the trial control limits
3.Data points outside the trial limits should be
investigated. Those points can only be removed if the
assignable causes can be detected and removed
4.Repeat steps 2 and 3 until no further action can be
taken
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)
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
.0027.
–
–
–
–
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
Table 4.1 Probabilities of Points
Plotting Outside Control Limits
n
1
2
5
10
15
20
25
50
100
350
.0027n
0.0027
0.0054
0.0135
0.0270
0.0405
0.0540
0.0675
0.1350
0.2700
0.9450
Actual Prob.
0.0027
0.0054
0.0134
0.0267
0.0397
0.0526
0.0654
0.1264
0.2369
0.6118
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)
4.3 Control Charts: When to Use,
Where to Use, How many to Use
Where:
• 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
4.4 Benefits from the Use of
Control Charts
• Good record keeping
• Control charts work as an aid in identifying special causes
of variation
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
4.6 Basic Statistical Aspects of
Control Charts
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.
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
Table 4.2 Data in Subgroups
Obtained at Regular Intervals
Subgroup
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X1
72
56
55
44
97
83
47
88
57
13
26
46
49
71
71
67
55
49
72
61
X2
84
87
73
80
26
89
66
50
47
10
39
27
62
63
58
69
63
51
80
74
X3
79
33
22
54
48
91
53
84
41
30
52
63
78
82
69
70
72
55
61
62
X4
49
42
60
74
58
62
58
69
46
32
48
34
87
55
70
94
49
76
59
57
X-bar
71.00
54.50
52.50
63.00
57.25
81.25
56.00
72.75
47.75
21.25
41.25
42.50
69.00
67.75
67.00
75.00
59.75
57.75
68.00
63.50
R
35
54
51
36
71
29
19
38
16
22
26
36
38
27
13
27
23
27
21
17
S
15.47
23.64
21.70
16.85
29.68
13.28
8.04
17.23
6.70
11.35
11.53
15.76
16.87
11.53
6.06
12.73
9.98
12.42
9.83
7.33
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)
Estimating of Population Parameters
by Sample Statistics
• Population Statistics:  , usually unknown
• Using Sample Statistics to estimate population
statistics:
R
– Point estimates
x

̂  x 
k
ˆ 
d2
s
ˆ 
c4
4.7.1 R-Chart
Figure 4.2 R-Chart
4.7.2 R-Chart with Probability Limits
4.7.3 S-Chart
S-Chart
4.7.4 S-Chart with Probability Limits
Table 4.3 The .001 and .999
Percentage Points of 2 Distribution
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1.5708E-06
0.0020
0.0243
0.0908
0.2102
0.3811
0.5985
0.8571
1.1519
1.4787
1.8339
2.2142
2.6172
3.0407
10.8276
13.8155
16.2662
18.4668
20.5150
22.4577
24.3219
26.1245
27.8772
29.5883
31.2641
32.9095
34.5282
36.1233
S-Chart with Probability Limits
4.7.5 S2-Chart
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?
4.7.7 Recomputing Control Limits
4.7.8 Applying Control Limits to
Future Production