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## Control charts

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### k. Statistically in control vs technically in control. statistically ... k. Shewhart control chart ... k. Range chart. need to monitor both mean and variance ... – PowerPoint PPT presentation

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Title: Control charts

1
Control charts
• 2WS02 Industrial Statistics
• A. Di Bucchianico

2
Goals of this lecture
• Further discussion of control charts
• variable charts
• Shewhart charts
• rational subgrouping
• runs rules
• performance
• CUSUM charts
• EWMA charts
• attribute charts (c, p and np charts)
• special charts (tool wear charts, short-run
charts)

3
Statistically versus technically in control
• Statistically in control
• stable over time /
• predictable
• Technically in control
• within specifications

4
Statistically in control vs technically in control
• statistically controlled process
• inhibits only natural random fluctuations (common
causes)
• is stable
• is predictable
• may yield products out of specification
• technically controlled process
• presently yields products within specification
• need not be stable nor predictable

5
Shewhart control chart
• graphical display of product characteristic which
is important for product quality

Upper Control Limit
Centre Line
Lower Control Limit
6
Control charts
7
Basic principles
• take samples and compute statistic
• if statistic falls above UCL or below LCL, then
out-of-control signal

how to choose control limits?
8
Meaning of control limits
• limits at 3 x standard deviation of plotted
statistic
• basic example

UCL
LCL
9
Example
• diameters of piston rings
• process mean 74 mm
• process standard deviation 0.01 mm
• measurements via repeated samples of 5 rings
yields

10
Individual versus mean
group means
individual observations
11
Range chart
• need to monitor both mean and variance
• traditionally use range to monitor variance
• chart may also be based on S or S2
• for normal distribution
• E R d2 E S (Hartleys constant)
• tables exist
• preferred practice
• first check range chart for violations of control
limits
• then check mean chart

12
Design control chart
• sample size
• larger sample size leads to faster detection
• setting control limits
• time between samples
• sample frequently few items or
• sample infrequently many items?
• choice of measurement

13
Rational subgroups
• how must samples be chosen?
• choose sample size frequency such that if a
special cause occurs
• between-subgroup variation is maximal
• within-subgroup variation is minimal.

14
Strategy 1
• leads to accurate estimate of ?
• maximises between-subgroup variation
• minimises within-subgroup variation

process mean
15
Strategy 2
• detects contrary to strategy 1 also temporary
changes of process mean

process mean
16
Phase I (Initial study) in control (1)
17
Phase I (Initial study) in control (2)
18
Phase I (Initial Study) not in-control
19
Trial versus control
• if process needs to be started and no relevant
historic data is available, then estimate µ and ?
or R from data (trial or initial study)
• if points fall outside the control limits, then
possibly revise control limits after inspection.
Look for patterns!
• if relevant historical data on µ and ? or R are
available, then use these data (control to
standard)

20
Control chart patterns (1)
• Cyclic pattern,
• three arrows with different weight

21
Control chart patterns (2)
• Trend,
• course of pin

22
Control chart patterns (3)
• Shifted mean,

23
Control chart patterns (4)
• A pattern can give explanation of the cause
• Cyclic ? different arrows, different weight
• Trend ? course of pin
• Shifted mean ? adjusted height Dartec
• Assumption a cause can be verified by a pattern
• The feather of one arrow is damaged ? outliers
below

24
Phase II Control to standard (1)
25
Phase II Control to standard (2)
26
Runs and zone rules
• if observations fall within control limits, then
process may still be statistically
out-of-control
• patterns (runs, cyclic behaviour) may indicate
special causes
• observations do not fill up space between control
limits
• extra rules to speed up detection of special
causes
• Western Electric Handbook rules
• 1 point outside 3?-limits
• 2 out of 3 consecutive points outside 2 ? -limits
• 4 out of 5 consecutive points outside 1 ? -limits
• 8 consecutive points on one side of centre line
• too many rules leads to too high false alarm rate

27
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28
Warning limits
• crossing 3 ? -limits yields alarm
• sometimes warning limits by adding 2 ? -limits
no alarm but collecting extra information by
• adjustment time between taking samples and/or
• warning limits increase detection performance of
control chart

29
Detection meter stick production
• mean 1000 mm, standard deviation 0.2 mm
• mean shifts from 1000 mm to 0.3 mm?
• how long does it take before control chart
signals?

30
Performance of control charts
• expressed in terms of time to alarm (run length)
• two types
• in-control run length
• out-of-control run length

31
Statistical control and control charts
• statistical control observations
• are normally distributed with mean ? and
variance ?2
• are independent
• out of (statistical) control
• change in probability distribution
• observation within control limits
• process is considered to be in control
• observation beyond control limits
• process is considered to be out-of-control

32
In-control run length
• process is in statistical control
• small probability that process will go beyond 3 ?
limits (in spite of being in control) - false
alarm!
• run length is first time that process goes beyond
3 ? limits
• compare with type I error

33
Out-of-control run length
• process is not in statistical control
• increased probability that process will go beyond
3 ? limits (in spite of being in control) - true
alarm!
• run length is first time that process goes beyond
3 sigma limits
• until control charts signals, we make type II
errors

34
Metrics for run lengths
• run lengths are random variables
• ARL Average Run Length
• SRL Standard Deviation of Run Length

35
Run lengths for Shewhart Xbar- chart
• in-control p 0.0027
• time to alarm follows geometric distribution
• mean 1/p 370.4
• standard deviation (?(1-p))/p 369.9

36
Geometric distribution
37
Numerical values
• Shewhart chart for mean (n1)
• single shift of mean

38
Scale in Statgraphics
• Are our calculations wrong???

39
Sample size and run lengths
• increase of sample size corresponding control
limits
• same in-control run length
• decrease of out-of-control run length

40
Numerical values
• Shewhart chart for mean (n5)
• single change of standard deviation (? - c?)

41
Runs rules and run lengths
• in-control run length decreases (why?)
• out-of-control run length decreases (why?)

42
Performance Shewhart chart
• in-control run length OK
• out-of-control run length
• OK for shifts 2 standard deviation group
average
• extra run tests
• decrease in-control length
• decrease out-of-control length

43
CUSUM Chart
• plot cumulative sums of observation

44
CUSUM tabular form
• assume
• data normally distributed with known ?
• individual observations

45
Choice K and H
• K is reference value (allowance, slack value)
• C measures cumulative upward deviations of µ0K
• C- measures cumulative downward deviations of
µ0-K
• for fast detection of change process mean µ1
• K½ µ0- µ1
• H5? is good choice

46
UCL
CL
change point
LCL
47
• only for two-sided schemes
• range of arms V-mask unclear
• interpretation parameters (angle, ...) not well
determined

48
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49
Rational subgroups and CUSUM
• extension to samples
• replace ? by ?/?n
• contrary to Shewhart chart , CUSUM works best
with individuals

50
Combination
• CUSUM charts appropriate for small shifts (
• CUSUM charts are inferior to Shewhart charts for
large shifts(1.5?)
• use both charts simultaneously with 3.5?
control limits for Shewhart chart

51
• increase detection power by restart process
• esp. useful when process mean at restart is not
equal at target value
• set C0 and C-0 to non-zero value (often H/2 )
• if process equals target value µ0 is, then CUSUMs
• if process mean does not equal target value µ0,
then faster alarm

52
CUSUM for variability
• define Yi (Xi-µ0)/ ? (standardise)
• define Vi (?Yi-0.822)/0.349
• CUSUMs for variability are

53
Exponentially Weighted Moving Average chart
• good alternative for Shewhart charts in case of
small shifts of mean
• performs almost as good as CUSUM
• mostly used for individual observations (like
CUSUM)
• is rather insensitive to non-normality

54
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55
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56
Why control charts for attribute data
• to view process/product across several
characteristics
• for characteristics that are logically defined on
a classification scale of measure
• N.B. Use variable charts whenever possible!

57
Control charts for attributes
Attributes are characteristics which have a
countable number of possible outcomes.
• Three widely used control charts for attributes
• p-chart fraction non-conforming items
• c-chart number of non-conforming items
• u-chart number of non-conforming items per unit
• For attributes one chart only suffices (why?).

58
p-chart
Number of nonconforming products is binomially
distributed
sample fraction of nonconforming
mean
variance
59
p-chart
average of sample fractions
Fraction Nonconforming Control Chart
60
Assumptions for p chart
• item is defect or not defect (conforming or
non-conforming)
• each experiment consists of n repeated
trials/units
• probability p of non-conformance is constant
• trials are independent of each other

61
c-chart
• Counts the number of non-conformities in sample.
• Each non-conforming item contains at least one
non-conformity (cf. p chart).
• Each sample must have comparable opportunities
for non-conformities
• Based on Poisson distribution
• Prob( nonconf. k)

62
c-chart
• Poisson distribution meanc and variancec

Control Limits for Nonconformities
is average number of nonconformities in sample
63
u-chart
• monitors number of non-conformities per unit.
• n is number of inspected units per sample
• c is total number of non-conformities

Control Chart for Average Number of
Non-conformities Per Unit
64
Moving Range Chart
• use when sample size is 1
• indication of spread moving range
• Situations
• automated inspection of all units
• low production rate
• expensive measurements
• repeated measurements differ only because of
laboratory error

65
Moving Range Chart
• calculation of moving range
• d2, D3 and D4 are constants depending number of
observations

individual measurements
moving range
66
Example Viscosity of Aircraft Primer Paint
67
Viscosity of Aircraft Primer Paint
• since a moving range is calculated of n2
observations, d21.128, D30 and D43.267

CC for individuals
CC for moving range
68
Viscosity of Aircraft Primer Paint
X
MR
69
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70
Tool wear chart
• known trend is removed (regression)
• trend is allowed until maximum
• slanted control limits

USL
UCL
reset
LCL
LSL
71
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72
Pitfalls