Statistical Process Control Charts

1
Statistical Process Control Charts

• Module 5

2
Goal

• monitor behaviour of process using measurements
  in order to determine whether process operation
  is statistically stable
• stable
• properties not changing in time
• mean
• variance
• on target?

3
Approach

• essentially a graphical form of hypothesis test
• use observed value of test statistic, and compare
  to limits at a desired confidence level
• limits reflect background variability in process
• common cause variation
• if a significant point is detected (hypothesis
  that nothing has changed is rejected), stop
  process and look for assignable causes
• special cause variation

4
X-bar Charts

• are used for testing stability of the mean
  operation
• calculate averages at regular intervals in time
  from samples of n elements taken at each time
  step
• centre-line - determined from
• target or specification value (to monitor whether
  we are on-specification
• reference data set - average of sample averages
  for data set collected when process was stable

N
1

X
X
å
j
N

j
1
5
X-bar Charts

• Control Limits - determined using estimate of
  dispersion (variability) in process
• 1. Using Estimated Standard Deviation
• from reference data set
• where sj is the sample standard devn. for the
  jth sample in the reference data set

N
1

å
s
s
j
N

j
1
6
X-bar Charts

• The sample standard devn provides a biased
  estimate of the true standard devn
• Thus, use the following as an estimate
• The control limits are

s
E
s
c
(
)
4
s
s

c
4
s
3
-

centre
line
c
n
4
Û
-

centre
line
A
s
3
3

with
A
3
c
n
4
7
X-bar Charts

• 2. Using Range
• estimate from reference data set
• for normally distributed data, range can be
  related to standard devn as follows -
• control chart limits

N
1

R
R
å
j
N

j
1
R
s

d
2
R
3
-

centre
line
d
n
2
3
Û
-


centre
line
A
R
with
A
2
2
d
n
2
8
X-bar Charts
significant change? look for assignable cause

UCL


centre- line




LCL
sample number (or time)
9
R-Charts

• Monitor range to determine whether variability is
  stable.
• Range provides an indication of dispersion, and
  is easy to calculate.
• calculate range at regular intervals from samples
  of n elements
• plot on chart with centre-line and control limits
• centre-line - from reference data set, computed
  as average of observed sample ranges

N
1

R
R
å
j
N

j
1
10
R-Charts

• control limits - determined by looking at
  sampling properties of range computed from
  observations
• control chart limits
• these limits are at the 99.7 confidence level
  (3-sigma limits for range)

UCL
D
R
4

LCL
D
R
3
11
s-Charts

• Monitor standard deviation to determine whether
  variability is stable - standard devn provides
  an indication of dispersion.
• calculate s at regular intervals from samples
  of n elements
• plot on chart with centre-line and control limits
• centre-line - from reference data set, computed
  as average of observed sample values

N
1

s
s
å
j
N

j
1
12
s-Charts

• control chart limits - computed by considering
  sampling distribution for sample standard devn
  s
• chart limits at 99.73 confidence level are -
  
• define

s
E
s
c


4
2
2

-
s
Var
s
c
(
)
(
)
1
4
æ
ö
2
-
c
1
ç

4

s
s
3
ç

c
è
ø
4
2
-
c
3
1
4

-
B
1
3
c
4
2
-
c
3
1
4


B
1
4
c
4
13
s-Charts

• chart limits are

B
s
3
B
s
4
14
Moving Range Charts

• How can we measure dispersion when we collect
  only one data point per sample?
• Answer - using the moving range - difference
  between adjacent sample values
• Use this approach to -
• obtain measure of dispersion for x-bar chart
  limits
• monitor consistency of variation in the process -
  MR-chart

-
MR
X
X
-
j
j
1
15
Using Moving Ranges for X-bar Limits

• Calculate average moving range from reference
  data set
• Convert AMR into an estimate for the standard
  deviation using the constant d2 for n2 sample
  points
• centre-line - use either a target value, or the
  average of the samples in the reference data set

N
1


-
å
AMR
M
R
X
X
-
j
j
1
-
N
1

j
2
M
R
-

centre
line
3
1
128
.
16
Monitoring Dispersion with Moving Ranges

• Use MR to monitor variability if you are only
  collecting one point per sample
• use average moving range from the reference data
  set as the centre-line
• chart limits are -
• upper limit D3 AMR
• lower limit D4 AMR

17
Tuning the SPC Chart

• The control limits and stopping rules influence
• false alarm rates - signal that a change has
  occurred when in fact it hasnt
• type I error - from hypothesis testing
• failure to detect rates - we dont recognize
  that a change has occurred when in fact it has
• type II error - from hypothesis testing
• When the number of data points per sample is
  fixed, there is a trade-off between false alarm
  and failure to detect rates.

18
Stopping Rules for Shewhart Charts

• Simplest stopping rule -
• alarm and stop when one of the measured
  characteristics exceeds the upper or lower
  control limit
• look for assignable causes
• false alarm rate - alpha - type I error
  probability
• failure to detect rate - beta - type II error
  probability
• We can conduct numerical simulation experiments
  (Monte Carlo simulations) to identify -
• how long, on average, it takes to detect a shift
  after it has occurred
• how long, on average, it takes before we receive
  a false alarm when no shift has occurred

19
Average Run Length (ARL)

• average time until a shift of a specified size
  is detected
• shift specified in terms of standard devn of the
  charted characteristic - to eliminate scale
  effects
• ARL(0)
• average time until false alarm occurs (no shift
  has occurred)
• ARL(1)
• average time until a shift of 1 standard devn in
  the charted characteristic is detected (e.g.,
  for sample average - shift of )

s
/
n
20
Stopping Rules

• Simple stopping rules may lead to unacceptable
  false alarm rates, or failure to detect modest
  shifts
• We can modify the rules to address these
  short-comings - for example, look for
• consecutive points above or below a reference
  line (e.g., two standard devns.)
• cyclic patterns
• linear trends
• One such set of guidelines are known as the
  Western Electric Stopping Rules.

unacceptable ARLs
21
Western Electric Stopping Rules

• 1) Stop if 2 out of 3 consecutive points are on
  the same side of the centre line, and more than 2
  std. devns from certain (warning lines)

upper control limit



s
2
X

centre line
22
Western Electric Stopping Rules

• 2) 4 out of 5 consecutive points lie on one side
  of the centre line, and are more than 1 standard
  devn from the centre line

upper control limit






s
1

X
centre line

23
Western Electric Stopping Rules

• 3) 8 consecutive points occurring on one side of
  the centre line

upper control limit








centre line

24
Western Electric Stopping Rules

• Stop if one of the following Trend Patterns
  occur

upper control limit
7 consecutive rising points (or falling points)






centre line


25
Western Electric Stopping Rules

• Trend Patterns -
• cyclic patterns - cycling about the centre-line
• periodic influence present in process?
• clustering pattern near centre-line
• sudden decrease in variance?
• clustering near the control limits -
• near the high limit
• near the low limit
• suggests two populations present in data - two
  distributions lying in the data
• effect of two processing paths, two types of
  feed, day vs. night shift?

26
Cusum Charts

• Cusum - cumulative sum
• Shewhart charts are effective at detecting major
  shifts in process operation.
• The goal of Cusum charts is more rapid detection
  of modest shifts in operation.
• Scenario - sustained shift, which is not large
  enough to exceed Shewhart chart limits - is
  something happening in the process?

27
Cusum Charts

• Approach - look at cumulative departures of the
  measured quantity (average, standard devn) from
  the target value
• For automated detection, keep two running totals
• Initialize these sums at 0.

i
(
)
-

å
target
X
S
j
i

j
1

-


))
(
(
,
0
max
k
target
X
SU
SU
-
1
i
i
i
-
-
-

))
(
(
,
0
max
k
target
X
SL
SL
-
1
i
i
i
28
Tuning the Cusum Chart

• Constant k - typically chosen as D/2, where D
  is the magnitude of the shift to be detected
• Chart Limit -
• where

2
-
s
b
æ
ö
1

ç

H
ln
è
ø
a
D
2
/
2
s

Var
X
(
)
a

probability of type I error - false alarm rate
b

probability of type II error - failure to detect
rate
29
Tuning the Cusum Chart

• reducing false alarm rate (alpha) leads to
  increase in chart limit - move the fence higher
• increased process variability leads to higher
  chart limit

30
Detecting Change

• Shewhart Charts
• look at current values
• useful for detecting major changes
• Cusum Charts
• look at complete history of values - cumulative
  sum
• all values are treated equally
• useful for detecting modest shifts
• Is there a compromise between these extremes?

31
EWMA Charts

• Exponentially Weighted Moving Average
• Use a moving average which weights recent values
  more heavily than older values
• more limited memory
• memory is adjustable via the weighting factor
• Exponentially Weighted Moving Average

-


l
l
E
x
E
)
1
(
-
t
t
t
1

target
E
weighting factor
0
32
Properties of EWMAs

• To see exponential weighting, consider
• Common values for weighting are
• however the weighting factor can be any value
  between 0 and 1.
• Large weighting factor short memory.

2
3


-

-

-

l
l
l
l
l
l
l
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x
x
x
x
(
)
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)
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2
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.
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3





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2
3


l
0
1
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3
.
.
33
Properties of EWMAs

• For a charted characteristic,
• Mean
• Variance
• where are properties of the
  characteristic being charted. For example, if we
  are charting the sample average,

m
E
E


t
2
s
l

Var
E
(
)
t
-
l
2
m
m
m


X
X
s
2
2
X
s
s


X
n
34
EWMA Control Limits

• Using the statistical properties of the EWMAs,
  choose control chart limits as

l

target
3
s
-
l
2
s
X

with
s
for
charting
sample
averages
n

sample
standard
devn.
of
process
s
X
35
Why do we need EWMA type charts?

• Shewhart Charts assume process is
• in particular, common cause variation at one
  sample time is independent of the variation at
  another time
• But what if it isnt?
• obtain misleading indication of process variance
• mean and/or variance may appear to wander when in
  fact they havent changed

m
e

mean
normally distributed random noise with zero
mean, constant variance
36
Why do we need EWMA-type charts?

• EWMA type charts account for possible
  dependencies between the random components
  (common cause variation) in the data, and are
  thus more representative.
• Causes of time dependencies in the common cause
  variation -
• inertia of process - fluctuations enter process
  and work their way through the process
• e.g., fluctuations entering the waffle batter
  mixing tank
• drifting in sensors - measurements for samples
  have a component which is wandering
• e.g., analytical equipment which requires
  re-calibration

37
Process Capability

• can be defined using concepts from Normal
  distribution
• Concept - compare specification limits to
  statistical variation in process
• apply to process whose statistical
  characteristics are stable
• Question
• do the range of inherent process variation lie
  within the specification limits?
• if specification limits are smaller, then we can
  expect to have more defects - values lying
  outside spec limits

38
Process Capability

• Specification limits
• operate between lower specification limit (LSL)
  and upper specification limit (USL)
• Statistical variation
• 99.73 of values for Normal distribution are
  contained in /- 3? ? 99.73 of values lie in
  interval of width 6?
• Cp
• defined as

-
LSL
USL

C
p
s
6
39
Process Capability

• Interpretation
• process capability lt 1 implies that specification
  limits are smaller than range of inherent
  variation
• process is NOT CAPABLE of meeting specifications
• Cp value of 1.3 - 1.4 indicates process is
  capable of meeting specifications a sufficiently
  large proportion of time

40
Process Capability

• Capability Index Cpk
• previous definition of Cp implies that operation
  is on target -- mean specification value -- so
  that specification interval and statistical
  variation intervals are centred at same point
• if mean operating point is closer to one of the
  specification limits, we can expect more defects
  due to statistical variation -- Cp provides
  misleading indication in this instance
• solution - compare distance between mean and spec
  limit to 3?, for each spec limit and select
  whichever is smaller

41
Process Capability

• Capability Index Cpk
• definition
• maximum value of Cpk is Cp
• Example - measurements of top surface colour of
  49 pancakes
• sample average 46.75
• sample standard deviation s 3.50
• LSL 43, USL 53

-
-
ü
ì
LSL
x
x
USL

,
minimum
C
ý
í
pk
s
s
þ
î
3
3
42
Process Capability

• Example
• indices
• Interpretation - current performance is
  unacceptable, and process is not capable of
  meeting specifications.

-
-
)
43
53
(
LSL
USL



48
.
0
C
p
s
)
50
.
3
(
6
6
-
-
-
-
)
75
.
46
53
(
)
43
75
.
46
(
x
USL
LSL
x


)
,
min(
)
,
min(
C
pk
s
s
)
50
.
3
(
3
)
50
.
3
(
3
3
3

36
.
0