Arab Academy for Science, Technology
& Maritime Transport
Lecture Four
Control Charts for
Variables (continue )
‫جميع حقوق الطبع والنشر محفوظة لألكاديمية العربية‬
‫للعلوم والتكنولوجيا والنقل البحري‬
What if the Process is Out of
Control
 If points plot out of control, then the control limits
must be revised.
 Before revising, identify out of control points and
look for assignable causes.
 If assignable causes can be found, then discard the
point(s) and recalculate the control limits.
 If no assignable causes can be found then
* 1) either discard the point(s) as if an assignable cause
had been found or
* 2) retain the point(s) considering the trial control
limits as appropriate for current control.
Introduction to SQC
Productivity and Quality Institute
S-2
Revised Control Limits for X and R
 Revised center line for x and R control
charts
xnew
x  xd


m  md
Rnew
R  Rd


m  md
where xd  discarded subgroup means
md  number of discarded subgroups
Rd  discarded subgroup ranges
Introduction to SQC
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S-3
Revised Control Limits
(Cont.)
xo  xnew
Ro  R new
R
σo  o
d2
Where d2 is a factor from the table.
Introduction to SQC
Productivity and Quality Institute
S-4
Revised Control Limits (Cont.)
Revised x chart
UCLx  xo  A o
Revised R Chart
UCLR  D2 o
LCLx  xo  A o
LCLR  D1 o
Where A, D1 and D2 are factors from the table
corresponding to subgroup size n.
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5
Example 1
 The following data represent the depth of
keyway taken in millimeters. We wish to
establish statistical control of the process
using x and R charts. Twenty five samples,
each of size 4, have been taken when we
think the process is in control.
 The depth of keyway measurements data from
these samples are shown in the next table.
Introduction to SQC
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S-6
Subgroup Data for Depth
Sample
Number
Date
Time
1
23/12
Measurements
Average
Range
x1
x2
x3
x4
8:50
6.35
6.40
6.32
6.37
6.36
0.08
2
11:30
6.46
6.37
6.36
6.41
6.40
0.01
3
1:45
6.34
6.40
6.34
6.36
6.36
0.06
4
3:45
6.69
6.64
6.68
6.59
6.65
0.10
5
4:20
6.38
6.34
6.44
6.40
6.39
0.10
8:35
6.42
6.41
6.43
6.34
6.40
0.09
7
9:00
6.44
6.41
6.41
6.46
6.43
0.05
8
9:40
6.33
6.41
6.38
6.36
6.37
0.08
9
1:30
6.48
6.44
6.47
6.45
6.46
0.04
24
2:00
6.43
6.43
6.35
6.38
6.38
0.08
25
4:25
6.39
6.39
6.43
6.44
6.41
0.06
Sum
160.25
2.19
6
27/12
Ri
Comment
New, temporary
operator
…
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7
Trial Control Limits
x
•Also, the
chart is
out of control, and
need revision.
•R chart has to be
revised first, and
then
chart
x
Xbar-R Chart for depth of keyway
1
Sample M ean
6.6
1
6.5
U C L=6.4737
__
X=6.4099
6.4
LC L=6.3461
1
6.3
1
3
5
7
9
11
13
Sample
15
17
19
21
23
25
1
0.3
Sample Range
•Since the R chart
indicates that the
process variability
is out of control,
we may revise the
chart.
0.2
U C L=0.1998
0.1
_
R=0.0876
0.0
LC L=0
1
3
5
7
9
11
13
Sample
15
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17
19
21
23
25
8
Identifying Out of Control Points
Test Results for Xbar Chart of X1, ..., X4
TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 4, 16, 20
Sample 4 : Assignable cause; New temporary operator
Sample 16: Random Cause
Sample 20: Assignable cause; Bad material
Test Results for R Chart of X1, ..., X4
TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 18
Sample 18: Assignable cause; Damage oil line
Introduction to SQC
Productivity and Quality Institute
S-9
Revised R Chart

The revised center line is
Rnew
RR
2.19  0.30



 0.0788
d
m  md
25  1
R0  Rnew  0.0788
R0 0.0788
0 

 0.03827
d2
2.059

The value of d2, for sample of size n = 4 and using the
table equal
 d2 =2.059
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10
Revised Control Limits for R
Chart
Therefore, the Revised control limits for the
R chart are
UCLR  D2 0
 (4.698)(0.03827)  0.1797
LCLR  D1 0
 (0)(0.038)  0
Introduction to SQC
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S-11
Revised Control Chart for R
R Chart for depth of keyway
•We can revise the x
chart, to control the
process center.
0.25
Sample Range
•Now, the R chart is in
control at the stated
level. i.e the variability
of the process is within
control.
1
0.30
0.20
UCL=0.1797
0.15
0.10
_
R=0.0788
0.05
0.00
LCL=0
1
Introduction to SQC
3
5
7
9
11
13 15
Sample
Productivity and Quality Institute
17
19
21
23
25
S-12
Revised Xbar Chart

The revised center line is
xnew
x  x 160.25  (6.65  6.51)



 6.3951
d
m  md
25  2
x0  xnew  6.3951

The value of d2, for sample of size n = 4 and using the
table equal
 d2 =2.059
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13
Revised Control Limits for
Chart X
Therefore, the Revised control limits for the
x chart are
UCLx  x0  A 0
 6.395  (1.5  0.03827)
 6.4525
LCLx  x0  A 0
 6.395  (1.5  0.03827)
 6.3377
Introduction to SQC
Productivity and Quality Institute
S-14
Revised Control Chart Xfor
•Note that subgroup 9 is
out of control. The
cause responsible for
this signal should be
treated as random
cause.
Introduction to SQC
Xbar Chart for depth of keyway
1
6.65
6.60
6.55
Sample Mean
•The x chart is now in
control at the stated
level; center line and
control limits.
1
6.50
1
6.45
UCL=6.4525
6.40
_
_
X=6.3951
6.35
LCL=6.3377
6.30
1
3
5
7
9
11
13 15
Sample
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17
19
21
23
25
S-15
Revised Control Charts

Since both the revised x and R charts exhibit control, we would
conclude that the process is in control at the stated level.

Adopt the Revised control limits for use in on-line statistical
process control.

The capability analysis can now be conducted if the specifications
limits on the depth of keyway is known.
Introduction to SQC
Productivity and Quality Institute
S-16
Example 2
 Estimate the mean depth of the keyway
data?
 Estimate the process standard deviation?
Introduction to SQC
Productivity and Quality Institute
S-17
Solution

We may estimate the mean depth of the keyway data as:
ˆ  x  6.3951 mm

The process standard deviation may be estimated using
the following equation:
ˆ   0 
R0
 0.03827
d2
 Where the value of d2 for samples of size 4 is found from
the table.
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Productivity and Quality Institute
S-18
Example 3
 The depth of keyway is normally distributed
with mean 6.3951 mm and standard deviation
0.03827. The specification limits on the depth
are 6.40 ± 0.06 mm.
 Estimate the fraction of nonconforming keyway.
 Also, calculate the capability of the process.
Introduction to SQC
Productivity and Quality Institute
S-19
Solution
p  P( x  LSL)  P( x  USL)
 P( x  6.34)  P( x  6.46)
6.34  6.3951
x   6.46  6.3951
)  P(

)

0.03827

0.03827
 P( z  1.44)  P( z  1.70)
 P(
x

 0.0749  0.0446
 0.1195  11.95%
 That is, about 11.95% [119500 parts per million
(ppm)] of the piston rings produced will be outside
the specifications.
Introduction to SQC
Productivity and Quality Institute
S-20
Solution (Cont.)



Note that 6 spread of the process is the basic definition of process
capability.
Since  is usually unknown, we must replace it with an estimate.

.
We frequently use σ̂ as an estimate of , resulting in an estimate C p of Cp

USL  LSL
Cp 

6
6.46  6.34
0.12


 0.52
6(0.03827) 0.22962
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S-21
Capability Using Minitab
Process Capability of X1, ..., X4
LSL Target USL
Within
Ov erall
P rocess Data
LS L
6.34
Target
6.4
USL
6.46
S ample M ean
6.3951
S ample N
100
S tDev (Within)
0.03827
S tDev (O v erall) 0.0733946
P otential (Within) C apability
Cp
0.52
C P L 0.48
C P U 0.57
C pk 0.48
O v erall C apability
Pp
PPL
PPU
P pk
C pm
6.24
O bserv ed P erformance
P P M < LS L
60000.00
P P M > U S L 130000.00
P P M Total
190000.00
Introduction to SQC
6.32
6.40
E xp. Within P erformance
P P M < LS L
74966.23
PPM > USL
44957.59
P P M Total
119923.82
6.48
6.56
0.27
0.25
0.29
0.25
0.28
6.64
E xp. O v erall P erformance
P P M < LS L 226405.56
P P M > U S L 188277.51
P P M Total
414683.07
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S-22
Solution (Cont.)
 This implies that the “natural” tolerance
limits in the process (three-sigma above
and below the mean) are well outside the
lower and upper specification limits.
 Consequently, a very large number of
nonconforming keyways will be produced.
Introduction to SQC
Productivity and Quality Institute
S-23
Guidelines for the Design of X
and R
 Specify sample size, control limit width, and
frequency of sampling.
 if the main purpose of the x chart is to detect
moderate to large process shifts, then small sample
sizes are sufficient (n = 4, 5, or 6).
 if the main purpose of the x chart is to detect small
process shifts, larger sample sizes are needed (as
much as 15 to 25)…which is often
impractical…alternative types of control charts are
available for this situation…
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Guidelines for the Design of X
and R (Cont. )
 If increasing the sample size is not an
option, then sensitizing procedures (such as
warning limits) can be used to detect small
shifts…but this can result in increased false
alarms.
 R chart is insensitive to shifts in process
standard deviation. (the range method
becomes less effective as the sample size
increases) may want to use S chart.
Introduction to SQC
Productivity and Quality Institute
S-25
Allocating Sample Effort
 Choose a larger sample size and sample
less frequently? or, Choose a smaller
sample size and sample more
frequently?
 The method to use will depend on the
situation. In general, small frequent
samples are more desirable.
Introduction to SQC
Productivity and Quality Institute
S-26
Charts Based on Standard
Values


If the process mean and variance are known or can
be specified, then control limits can be developed
using these values:
x  chart
R  chart
UCL  μ  Aσ
UCL  D2σ
CL  μ
CL  d 2σ
LCL  μ  Aσ
LCL  D1σ
The constant A, d2, D1, and D2 are given in the table
for various values of n
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Control Charts for X and S
 First, S2 is an “unbiased” estimator of 2
 Second, S is NOT an unbiased estimator of


S is an unbiased estimator of c4  where c4
is a constant.
 The standard deviation of S is
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σ 1  c42
S-28
Construction and Operation of
X and S Control Chart
 If a standard  is given the control limits
for the x and S charts are:
UCLx  x  A
UCLS  B6
CL  x
CL  c4
LCL  x  A
LCLS  B5
 B5, B6, A, and c4 are found in the table for
various values of n.
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Construction and Operation of
X and S Control Chart
 If no standard is given (i.e  is
unknown), we can use an average sample
standard deviation,1 m
S
Si

m
i 1
 The upper and lower control limits of the
S chart are
UCLs  B4 S
CL  S
LCLs  B3 S
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S-30
Construction and Operation of
X and S Control Chart

The upper and lower control limits for the
are given as
chart
x
UCLx  x  A3 S
CL  x
LCLx  x  A3 S
 where A3, B4, and B3 are found in the table.
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S-31
Estimating Process Standard
Deviation
 The process standard deviation,  can be
estimated by
S
σˆ 
c4
 Where c4 is a constant found in the table for
various values of n
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Example : Piston Ring Data
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33
Solution
 The grand average and the average standard
deviations are
25
x
i
1850.028
x

 74.001 mm
25
25
and
i 1
25
S
S
i 1
Introduction to SQC
25
i
0.2350

 0.0094
25
Productivity and Quality Institute
S-34
Solution (Cont.)

The trial control limits for the x chart is
UCLx  x  A3 S  74.001  (1.427)(0.0094)  74.014
CL  x  74.001
LCLx  x  A3 S  74.001  (1.427)(0.0094)  73.988

And for the S chart is
UCLS  B4 S  (2.089)(0.0094)  0.0196
CL  S  0.0094
LCLS  B3 S  (0)(0.0094)  0
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S-35
Plot the Trial Control Limits
Xbar-S Chart for Piston Ring Data
U C L=74.01459
Sample M ean
74.01
_
_
X=74.00118
74.00
73.99
LC L=73.98776
1
3
5
7
9
11
13
Sample
15
17
19
21
23
25
Sample StDev
0.020
U C L=0.01964
0.015
_
S =0.00940
0.010
0.005
0.000
LC L=0
1
3
Introduction to SQC
5
7
9
11
13
Sample
15
17
19
21
Productivity and Quality Institute
23
25
S-36