ENGM 720 - Lecture 07
X-bar, R & S Control Charts;
ARL & OC Curves
4/8/2015
ENGM 720: Statistical Process Control
1
Assignment:

Reading:
•
•

Chapter 5
•
Start & Finish reading
Chapter 6
•
Spring Break means reading statistics on the beach, right?
Assignments:
•
•
Obtain the draft Ctrl Chart Factors table from Materials Page
Access Excel Template for X-bar, R, & S Control Charts:
•
•
Download Assignment 5 for practice
Use the Excel sheet to do the charting, and verify your hand
calculations
4/8/2015
ENGM 720: Statistical Process Control
2
Process for Statistical Control
Of Quality

Removing
special causes
of variation
Statistical Quality Control and Improvement
Improving Process Capability and Performance
• Hypothesis
Tests
• Ishikawa’s
Continually Improve the System
Characterize Stable Process Capability
Tools

Managing the
process with
control charts
Head Off Shifts in Location, Spread
• Process
Improvement
• Process
Stabilization
• Confidence in
Time
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
“When to Act”
Center the Process
LSL
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0
USL
ENGM 720: Statistical Process Control
3
Moving from Hypothesis Testing
to Control Charts

A control chart is like a sideways hypothesis test
• Detects a shift in the process
• Heads-off costly errors by detecting trends

2

2

2
0
2-Sided Hypothesis Test
4/8/2015
UCL
0
CL

2
Sideways Hypothesis
Test
LCL
Sample Number
Shewhart Control Chart
ENGM 720: Statistical Process Control
4
Test of Hypothesis

A statistical hypothesis is a statement about the value of
a parameter from a probability distribution.

Ex. Test of Hypothesis on the Mean

•
•
Say that a process is in-control if its’ mean is 0.
In a test of hypothesis, use a sample of data from the
process to see if it has a mean of 0 .
Formally stated:
•
•
H0:  = 0
HA:  ≠ 0
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(Process is in-control)
(Process is out-of-control)
ENGM 720: Statistical Process Control
5
Test of Hypothesis on Mean
(Variance Known)

State the Hypothesis
•
•
H0:  = 0
H1:  ≠ 0

Take random sample from process and compute appropriate test
statistic
x  0
x  0
z0 

x

n

Pick a Type I Error level () and find the critical value z/2
Reject H0 if |z0| > z/2

 2
 2
 z
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z
2
ENGM 720: Statistical Process Control
z
2
6
UCL and LCL are Equivalent to
the Test of Hypothesis

Reject H0 if:
•
Case 1:
z0 
x  0


n
 z
2
 x   0  z
Case 2:
2
x  0
0  x
 z

n

n
)  UCL

n
2
 x   0  z

2
x  0
x  0
•
n
 z
2
)  LCL
For 3-sigma limits z/2 = 3
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ENGM 720: Statistical Process Control
7
Two Types of Errors May Occur
When Testing a Hypothesis


Type I Error - 
•
•
Reject H0 when we shouldn't
Analogous to false alarm on control chart, i.e.,
•
point lays outside control limits but process is truly in-control
Type II Error - b
•
•
Fail to reject H0 when we should
Analogous to insensitivity of control chart to problems, i.e.,
•
point does not lay outside control limits but process is never-theless out-of-control
4/8/2015
ENGM 720: Statistical Process Control
8
Choice of Control Limits:
Trade-off Between Wide or Narrow
Control Limits

Moving limits further from the center line
•
Decreases risk of false alarm, BUT increases risk of insensitivity
x
UCL
CL
LCL
Sam ple

Moving limits closer to the center line
•
Decreases risk of insensitivity, BUT increases risk of false alarm
x
UCL
CL
LCL
Sam ple
4/8/2015
ENGM 720: Statistical Process Control
9
Consequences of Incorrect
Control Limits


Bad Thing 1:
•
A control chart that never finds anything wrong
with the process, but the process produces bad
product
Bad Thing 2:
•
Too many false alarms destroy the operating
personnel’s confidence in the control chart, and
they stop using it
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ENGM 720: Statistical Process Control
10
Differences in Viewpoint Between Test of
Hypothesis & Control Charts
Hypothesis Test
Checks for the validity of
assumptions.
(ex.: is the actual process mean
what we think it is?)
Tests for sustained shift
(ex.: have we actually reduced the
variation like we think we have?)
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Control Chart
Detect departures from
assumed state of statistical
control
Detects shifts that are
short-lived
Detects steady drifts
Detects trends
ENGM 720: Statistical Process
Control
11
Example: Part Dimension

When a process is in-control, a dimension is normally
distributed with mean 30 and std dev 1. Sample size is 5.
Find the control limits for an x-bar chart with a false alarm
rate of 0.0027.
•
r.v. x - dimension of part
x ~ N    30,
•
  1)
r.v. x - sample mean dimension of part

x ~ N   30,  X  
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n 1
ENGM 720: Statistical Process Control
5
)
12
Distribution of x vs. Distribution of x
Distribution of
individual
measurements x :
N   , )
Distribution of
sample mean x :

N  , x  
n
)
UCL    3 x
CL  
LCL    3 x
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ENGM 720: Statistical Process Control
13
Ex. Part Dimension
Cont'd

Find UCL:
P  x  UCL   0.0027 / 2  0.00135

UCL   
P z 
  0.00135

n



UCL   
P z 
  1  0.00135  0.99865
 n 



UCL  
3
 n

 UCL    3 
n
)
The control limits are:
CL    30

UCL    3 

LCL    3 
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)

n  30  3 1
)

n  30  3 1
ENGM 720: Statistical Process Control
)
5  31.34
)
5  28.66
14
Ex. Modified Part Limits

Consider an in-control process. A process
measurement has mean 30 and std dev 1 and n = 5.
•
Design a control chart with prob. of false alarm = 0.005
α  .005 
α
 .0025
2
1  .0025 .9975 z α  2.81
2
 σ 
 1 
UCL  μ  2.81
  30  2.81
  31.3
 n
 5
 σ 
 1 
LCL  μ  2.81
  30  2.81
  28.7
 n
 5
•
If the control limits are not 3-Sigma, they are called
"probability limits".
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ENGM 720: Statistical Process
Control
15
General Model:
Shewhart Control Chart

Suppose x is some quality characteristic, and w is a sample
statistic of x.

Suppose mean of w is μw and std dev of w is σw, then:
•

•
•
UCL = μ w + Lσw
•
LCL = μ w – Lσw
CL = μ w
where L is the “distance” of the control limits from the center
line, and expressed in multiples (units) of the standard
deviation of the statistic, i.e. σw.
This type of chart is called a Shewhart Control Chart
4/8/2015
ENGM 720: Statistical Process Control
16
Rational Subgroups

Subgroups / Samples should be selected so that if
assignable causes are present:
•
•
Chance for differences between samples is maximized
Chance for differences within a sample is minimized

Use consecutive units of production

Keep sample size small so that:
•
•
•
New events won’t occur during sampling
Inspection is not too expensive
But size is large enough that x is normally distributed
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ENGM 720: Statistical Process Control
17
Why Monitor Both Process Mean and
Process Variability?
Process Over Time
Process Doing OK
Lower
Specification
Limit
Control Charts
Upper
Specification
Limit
X-bar
R
X-bar
R
X-bar
R
Mean shift in process
Increase in process variance
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ENGM 720: Statistical Process Control
18
Teminology

Causes of Variation:
•
Assignable Causes
Meaning of Control:
•
• Keep the process from
•
•

operating predictably
Things that we can do
something about
Common / Chance
Causes
• Random, inherent
In Specification
• Meets customer
constraints on product
•
In Statistical Control
• No Assignable
Causes of variation
present in the process
variation in the process
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ENGM 720: Statistical Process Control
19
Statistical Basis of x Chart


Suppose a quality characteristic is x ~ N(, )
and we know  and 
If x1, x2, …, xn is a random sample of size n then:
and x 
n
1
n
x
i 1
x ~ N

i
 ,
x 
n
)
Recall that the probability is  that either:
x  UCL    z
or
x  LCL    z
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
2
n

2
n
ENGM 720: Statistical Process Control
20
Statistical Basis of x Chart
Cont'd

which is equivalent to:

 

P    z 2
 x    z 2
 1

n
n

P  LCL  x  UCL   1  
•

Where LCL and UCL are the lower and upper control limits,
respectively
In practice, one must estimate  and  from data coming from
an in-control process
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ENGM 720: Statistical Process Control
21
Statistics of the Range


R – the range – is a sample statistic
If x1, x2, …, xn is a random sample of size n from a normal
distribution then one can estimate  using the range:
ˆ  R d 2
•

where d2 is a function of n and can be found in Appendix VI
Can get a better estimate for  if using more than one sample
•
•
Compute Ri for each of m samples where i = 1, …, m
Then use the sample average of Ri
m
R   Ri
i 1
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ˆ  R d2
ENGM 720: Statistical Process Control
22
Computing Trial Control Limits
for x Chart

Assume a quality characteristic x ~ N(, )

Take m  20 samples of size n = 4, 5, or 6

For each sample i, compute x and Ri for i = 1, …, m

Compute: x and R
m
x   xi
i 1
4/8/2015
m
R   Ri
i 1
ENGM 720: Statistical Process Control
23
Computing Trial Control Limits
for x Chart

General model for x chart
UCL   x  L x
CL   x

LCL   x  L x
Substituting estimates for μx and σx and using 3-sigma limits:
CL   x  x
  
UCL   x  3 x  x  3 

 n

 R d2 
 x  3
  x  A2 R
n 

Where A2 comes from Appendix VI and depends on n
LCL  x  A2 R
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ENGM 720: Statistical Process Control
24
Computing Trial Control Limits
for R - Chart

x and R charts come as a pair

General model for R chart
UCL   R  L R
CL   R
LCL   R  L R

Substituting estimates for R and R and using 3-sigma limits
CL   R  R
UCL   R  3 R  R  3  d 3 )
 R  3d 3
R
 D4 R
d2
LCL  D3 R
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ENGM 720: Statistical Process Control
25
Computing Trial Control Limits
for R - Chart (continued)

where
and
d3
D3  1  3
d2
d3
D4  1  3
d2

D3 and D4 are tabulated in Appendix VI and depend on n

NOTE: R chart is quite sensitive to departures from normality
4/8/2015
ENGM 720: Statistical Process Control
26
Control Chart Factors Table
(Appendix VI – see Materials Page for Engineering Notebook Copy)

For a constant sample size (n) and 3σ limits:
Obs n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
A
2.121
1.732
1.500
1.342
1.225
1.134
1.061
1.000
0.949
0.905
0.866
0.832
0.802
0.775
0.750
0.728
0.707
0.688
0.671
0.655
0.640
0.626
0.612
0.600
A2
1.880
1.023
0.729
0.577
0.483
0.419
0.373
0.337
0.308
0.285
0.266
0.249
0.235
0.223
0.212
0.203
0.194
0.187
0.180
0.173
0.167
0.162
0.157
0.153
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A3
2.659
1.954
1.628
1.427
1.287
1.182
1.099
1.032
0.975
0.927
0.886
0.850
0.817
0.789
0.763
0.739
0.718
0.698
0.680
0.663
0.647
0.633
0.619
0.606
d2
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
3.173
3.258
3.336
3.407
3.472
3.532
3.588
3.640
3.689
3.735
3.778
3.819
3.858
3.895
3.931
d3
0.853
0.888
0.880
0.864
0.848
0.833
0.820
0.808
0.797
0.787
0.778
0.770
0.763
0.756
0.750
0.744
0.739
0.734
0.729
0.724
0.720
0.716
0.712
0.708
D1
0
0
0
0
0
0.204
0.388
0.547
0.687
0.811
0.922
1.025
1.118
1.203
1.282
1.356
1.424
1.487
1.549
1.605
1.659
1.710
1.759
1.806
D2
3.686
4.358
4.698
4.918
5.078
5.204
5.306
5.393
5.469
5.535
5.594
5.647
5.696
5.741
5.782
5.820
5.856
5.891
5.921
5.951
5.979
6.006
6.031
6.056
D3
0
0
0
0
0
0.076
0.136
0.184
0.223
0.256
0.283
0.307
0.328
0.347
0.363
0.378
0.391
0.403
0.415
0.425
0.434
0.443
0.451
0.459
D4
3.267
2.574
2.282
2.114
2.004
1.924
1.864
1.816
1.777
1.744
1.717
1.693
1.672
1.653
1.637
1.622
1.608
1.597
1.585
1.575
1.566
1.557
1.548
1.541
c4
0.7979
0.8862
0.9213
0.9400
0.9515
0.9594
0.9650
0.9693
0.9727
0.9754
0.9776
0.9794
0.9810
0.9823
0.9835
0.9845
0.9854
0.9862
0.9869
0.9876
0.9882
0.9887
0.9892
0.9896
B3
0
0
0
0
0.030
0.118
0.185
0.239
0.284
0.321
0.354
0.382
0.406
0.428
0.448
0.466
0.482
0.497
0.510
0.523
0.534
0.545
0.555
0.565
B4
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
1.679
1.646
1.618
1.594
1.572
1.552
1.534
1.518
1.503
1.490
1.477
1.466
1.455
1.445
1.435
Table factors derived from Montgomery,D.C., (2005) Statistical Quality Control, 5th Ed.
SHORT RUN SPC
B5
0
0
0
0
0.029
0.113
0.179
0.232
0.276
0.313
0.346
0.374
0.399
0.421
0.440
0.458
0.475
0.490
0.504
0.516
0.528
0.539
0.549
0.559
27
B6
2.606
2.276
2.088
1.964
1.874
1.806
1.751
1.707
1.669
1.637
1.610
1.585
1.563
1.544
1.526
1.511
1.496
1.483
1.470
1.459
1.448
1.438
1.429
1.420
Trial Control Chart Limits:
Guidelines for Sampling

Sample should be of size 3 to 8
(sizes 4 – 6 are more common)

Sample must be homogeneous
• same time (consecutive units)
• same raw materials
• same operator
• same machine

Time may pass between samples but not within samples
4/8/2015
ENGM 720: Statistical Process Control
28
Steps for Trial Control Limits




Start with 20 to 25 samples
Use all data to calculate initial control limits
Plot each sample in time-order on chart.
Check for out of control sample points
•
•
If one (or more) found, then:
1. Investigate the process;
2. Remove the special cause; and
3. Remove the special cause point and recalculate
control limits.
If can’t find special cause - drop point &
recalculate anyway
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ENGM 720: Statistical Process Control
29
Control Chart Sensitizing Rules


Western Electric Rules:
1.
One point plots outside the three-sigma limits;
2.
Eight consecutive points plot on one side of the center line;
3.
Two out of three consecutive points plot beyond two-sigma
warning limits on the same side of the center line; or
4.
Four out of five consecutive points plot beyond one-sigma
warning limits on the same side of the center line.
If chart shows lack of control, investigate for special
cause
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ENGM 720: Statistical Process Control
30
Control Chart Examples
UCL
UCL
UCL
UCL
x
x
x
x
LCL
LCL
LCL
LCL
Rule 1
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Rule 2
Rule 3
ENGM 720: Statistical Process Control
Rule 4
31
Control Chart Sensitizing Rules

Additional Sensitizing Rules:
One or more points very near a control limit.
6. Six points in a row steadily increasing or decreasing.
7. Eight points in a row on both sides of the center line, but
none in-between the one-sigma warning limits on both
sides of the center line.
8. Fourteen points in a row alternating above and below the
center line.
9. Fifteen points in a row anywhere between the one-sigma
warning limits (including either side of the center line).
10. Any unusual or non-random pattern to the plotted points.
5.
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ENGM 720: Statistical Process Control
32
Charts Based on
Standard Values, x Chart

If values for  and  are known
(i.e., do not need to estimate from data)
UCL   x  3 x
CL   x
LCL   x  3 x

n    A

n    A
UCL    3 
CL  
LCL    3 

)
)
Quantity A is tabulated in Appendix VI
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ENGM 720: Statistical Process Control
33
R - Chart Based on
Standard Values

If values for R and  are known
UCL = R  3 R
CL   R
LCL   R  3 R


r.v. W = R /  – relative range
The parameters of the distribution of W are a function of n
W  E W   d 2

From the relative range we can compute the mean of R
R  W
 R  E  R   E W    E W   d 2
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ENGM 720: Statistical Process Control
34
The Standard Deviation of R


The standard deviation of R is given as:
(Text does not derive this)
therefore
 R  d3
UCL  d 2  3d 3
CL  d 2
LCL  d 2  3d 3

where
•

UCL  D2
CL  d 2
LCL  D1
D1  d 2  3d3
D2  d 2  3d3
D1 and D2 are constants tabulated in Appendix VI
Caution: Be careful when using standard values
• make sure these values are representative of the actual process
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X-Bar & R-Charts
 The X-Bar Chart
 The R-Chart checks
checks variability in for changes in
location between
sample variation
samples
UCL
UCL
x
R
LCL
LCL
Sample Number
X-Bar ( Means ) Control Chart
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Sample Number
R - ( Range ) Control Chart
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X-Bar & Sigma-Charts
 Used
when sample size is greater than 10
 X-Bar
Control Limits:
 Sigma-Chart
Control
• Approximate 3 limits are Limits:
found from S & table
• Approximate, asymmetric
3 limits from S & table
UCL  x  A 3 S
UCL  B4 S
CL  x
CL  S
LCL  x  A 3 S
LCL  B3 S
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X-Bar & Sigma-Charts
Limits can
 X-Bar
also be generated from historical data:
Control Limits:
 Sigma-Chart
Control
• Approximate 3 limits are Limits:
found from known 0 &
• Approximate, asymmetric
3 limits from 0 & table
table
UCL  x  A
UCL  B 6 
CL  x
CL  c 4 
LCL  x  A
LCL  B 5 
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Operating Characteristic
(OC) Curve



Ability of the x and R charts to detect shifts (sensitivity)
is described by OC curves
For x chart; say we know 
•
Mean shifts from
0
(in-control value) to
1 = 0 +k (out-of-control value)
The probability of NOT detecting the shift on the first
sample after shift is
b  P LCL  x  UCL   1  0  k 
 LCL  1
UCL  1 
b  P
z

 n 
  n
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OC Curve for x Chart

Plot of b vs. shift size (in std dev units) for various sample
sizes n
OC Curve for x-bar chart with 3-sigma limits
1.00
0.90
0.80
Beta
0.70
0.60
0.50
n=20
n=5
0.40
n=2
n=1
0.30
0.20
0.10
4.
8
4.
4
4.
0
3.
6
3.
2
2.
8
2.
4
2.
0
1.
6
1.
2
0.
8
0.
4
0
0.00
k

x chart not effective for small shift sizes, i.e., k  1.5

Performance gets better for larger n and L or larger shifts
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OC curve for R Chart

Uses distribution of relative range r.v., i.e.,

Suppose

 0 - in-control std dev
 1 - out-of-control std dev
OC curve for R chart plots b vs. ratio of in-control to
out-of-control standard deviation for various sample
sizes
•

W R 
That is, plot β vs. l  1/0
R chart not very effective for detecting shifts for small
sample sizes (see Fig. 5-14 in text)
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Probability of Detecting Shift for
Subsequent Samples

After the shift has occurred:
•
•
•
•
•
•
P(NOT detecting shift ON 1st sample)
b  0.07078
P(DETECTING shift ON 1st sample)
1  b )  0.93
P(DETECTING shift ON 2nd sample)
b 1  b )  0.066
P(DETECTING shift ON rth sample)
b r 1 1  b )
P(DETECTING shift BY 2nd sample)
1  b )  b 1  b )  0.93  0.066  0.996
P(DETECTING shift BY rth sample)

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i 1
b
1  b )
i 1
r
ENGM 720: Statistical Process Control
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Average Run Length (ARL)

Expected number of samples taken before shift
is detected is called the Average Run Length
(ARL)

ARL   r b
r 1
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r 1
1
1  b ) 
1  b )
ENGM 720: Statistical Process Control
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Performance of Any Shewhart
Control Chart

In-Control ARL:
•
Average number of points plotted on control chart
before a false alarm occurs
(ideally, should be large)
ARL0 

1

Out-of-Control ARL:
•
Average number of points, after the process goes outof-control, before the control chart detects it
(ideally, should be small)
ARL1 
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1
1 b
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ARL Curve for x Chart

Plot of ARL1 vs. shift size (in sd units) for various
sample sizes n:
3.0
2.9
2.7
2.6
2.4
2.3
2.1
2.0
1.8
1.7
n=1
1.5
1.4
n=2
1.2
1.1
0.8
n=4
0.6
0.5
0.3
0.2
n=20
0.9
20.00
18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.0
ARL to detect shift
ARL for x-bar chart with 3-sigma limits
k (shift size)

Average Time to Signal, (ATS):
•
Number of time periods that occur until signal is generated
on control chart
ATS   ARL ) h )
•
h - time interval between samples
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Questions & Issues
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