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Lecture 4Lecture 4Control Chart for VariablesControl Chart for Variables
1
�� Control Chart for VariablesControl Chart for Variables
Introduction
• Variable - a single quality characteristic that can be measured on a numerical scale.
• When working with variables, we should
2
• When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic.
Introduction
� Mean = central tendency of a process� Variability = process dispersion
3
µ0 µ1LSL USL
IntroductionIntroduction
σ0
4
µ0LSL USL
σ1
IntroductionIntroduction
� Monitor Mean Quality Level� x-bar control chart
Monitor Process Variability Quality
5
� Monitor Process Variability Quality Level� Control chart for standard deviation, S
chart� Control chart for the range, R chart
Selection Char. for Investigation
� There can be many possible quality characteristics
� The decision making process becomes
6
� The decision making process becomes more complicated
� Selecting a few vital quality char. from the many candidate ( using Pareto analysis )
Construction of Control Chart
� Selection of Rational Subgroups� Dif. among subgroups : maximized� Dif. within subgroups : minimized
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� Dif. within subgroups : minimized
� Subgroup Size� Normally between 4 – 10 ( 4 or 5 )
� Frequency of Subgroups Selection� Type of Measuring Instrument� Design of Recording Form for Data
Construction of Control Chart
Notation for variables control charts� n - size of the sample (sometimes called a
subgroup) chosen at a point in time� m - number of samples selected
8
� m - number of samples selected� = average of the observations in the ith
sample (where i = 1, 2, ..., m)� = grand average or “average of the
averages (this value is used as the center line of the control chart)
ix
x
� Ri = range of the values in the ith sample
Ri = xmax - xmin
Construction of Control Chart
9
Ri = xmax - xmin
� = average range for all m samples� µ is the true process mean� σ is the true process standard
deviation
R
Control Chart for the Mean and Range
� Development of the Chart ( Electric Resistors )
Sample X1 X2 X3 X4 X5
10
Sample X1 X2 X3 X4 X5
1 102,29 101,51 98,22 97,42 103,55
2 103,48 100,17 100,64 102,25 98,11
… … … … … …
… … … … … …
54 97,6217 98,1163 98,9194 99,0108 102,3500
55 100,2375 98,5452 98,0346 101,3721 99,0354
Control Chart for the Mean and Range
� Development of the Chart
Rn
X
gg
n
i i
R
XXX
∑∑
∑ = −==minmax
1 ,
11
gR
gX
g
i i
g
i i RX ∑∑ == == 11 ,
( ) RAXLCLUCLXX 2, ±=
RDLCLRDUCL RR 34 , ==
Control Chart for the Mean and Range
� Development of the Chart
Sample X1 X5 Xbar R UCL CL LCL
1 102,2915 … 103,5511 100,5976 6,132277 100,9477 100,9317 100,9158
12
1 102,2915 … 103,5511 100,5976 6,132277 100,9477 100,9317 100,9158
2 103,4836 … 98,11246 100,9317 5,371116 100,9477 100,9317 100,9158
… … … … … … … … …
… … … … … … … … …
54 97,62171 … 102,35 99,20364 4,728281 100,9477 100,9317 100,9158
55 100,2375 … 99,03541 99,44496 3,33747 100,9477 100,9317 100,9158
Control Chart for the Mean and Range
� Development of the Chart
Xbar chart, Electrical Resistors data, n=5 , trial limits
103.00
104.00
13
95.00
96.00
97.00
98.00
99.00
100.00
101.00
102.00
103.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
sample number
Me
an r
es
ista
nce
(o
hm
s)
Xbar
UCL
CL
LCL
Control Chart for the Mean and Range
� Development of the ChartR chart, Electric Resistors data, n=5, m=5
9
10
14
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
sample number
Ran
ge
(o
hm
s)
UCL
CL
LCL
Ranges
Control Chart for the Mean and Range
� CL for given target or standard
3, 000 +== σ
nXUCLXCL
XX
15
deviation standard
mean process theof uetarget val
3
0
0
00
==
−=
σ
σ
X
nXLCL
n
X
Control Chart for the Mean and Standard Deviation
� No Given Standard� s chart
sBLCLsBUCLs
sCL
g
i i1 ,, ==== ∑ =
16
� X bar chart
sBLCLsBUCLg
ssCL ss
i is 34
1 ,, ==== ∑ =
sAXLCL
sAXUCLg
XXCL
X
X
g
ii
X
3
31 ,
−=
+=== ∑ =
Control Chart for the Mean and Standard Deviation
� Given Standard� s chart
050604 ,, σσσ BLCLBUCLCCL sss ===
17
� X bar chart
050604 ,, σσσ BLCLBUCLCCL sss ===
00
000
3
,
σ
σ
−=
+==
XLCL
AXUCLXCL
X
XX
Example
The thickness of magnetic coating on audio tape is an important
Samplenumber
Sample mean
sample stdv
1 36,4 4,6
2 35,8 3,7
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important characteristics. Table shows the mean and standard dev for 20 samples. The spec are 38 ± 4.5 .
2 35,8 3,7
3 37,3 5,2
18 39,2 4,8
19 36,8 4,7
20 37,7 5,4
Example� Find the trial control limit for an X and s Chart
79.420
8.95
20
20
1 ==== ∑ =s
sCL i i
s
19
0)79.4)(0(
854.10)79.4)(266.2(
79.42020
3
4
===
===
====
sBLCL
sBUCL
sCL
s
s
s
Example� Find the trial control limit for an X and s Chart
075.3720
5.741
20
20
1 ==== ∑ =X
XCL ii
X
20
277.29)79.4)(628.1(075.37
873.44)79.4)(628.1(075.37
2020
3
3
=−=−=
=+=+=
sAXLCL
sAXUCL
X
X
X
Example : S chart
4
5
6
7
CL
21
0
1
2
3
4
1 3 5 7 9 11 13 15 17 19 LCL
Example : X Chart
37383940
CL
22
313233343536
1 3 5 7 9 11 13 15 17 19
Example� Assuming the thickness to be normally dist. what
proportion of the product will not meet spec ?
199.59213.0
79.4ˆ ===
c
sσ
23
1492.004.1199.5
075.375.42
2451.069.0199.5
075.375.33
199.59213.0
1
1
4
⇒=−=
⇒−=−=
===
z
z
cσ
proportion of the product not meet spec = 0.2451 + 0.1492 = 0.3943
Control Chart for Individual Units
� Variability of the process is estimated from the moving range ( subtracting the lesser value )
� No Given Standard
24
RDLCLRDUCLRCL RRR 34 ,, ===
22
3,3d
RXLCL
d
RXUCL
XCL
XX
X
−=+=
=
Control Chart for Individual Units
� Given Standard
02
, σσσ
dDLCLdDUCL
dCLR
===
25
023024 , σσ dDLCLdDUCL RR ==
0000
0
3,3 σσ −=+=
=
XLCLXUCL
XCL
XX
X
Example
Sample number
BrinellHardness
Moving range
1 36,3 0
The table shows the brinell hardness numbers of 20 individual stell
26
2 28,6 7,7
3 32,5 3,9
individual stell fasteners. Construct the X chart and moving range chart
18 36,2 7,7
19 30 6,2
20 28,3 1,7
Example
053.519
96
19
19
1 ==== ∑ RRCL i
R
27
0
508.16)053.5)(267.3(
1919
3
4
==
===
RDLCL
RDUCL
R
R
Example
20
1 8.3220
656
20==== ∑ i
X
XXCL
28
0
2020
361.19128.1
053.538.323
239.46128.1
053.538.323
2
2
=−=−=
=+=+=
d
RXLCL
d
RXUCL
X
X
Example : Moving range chart
8
10
12
29
0
2
4
6
8
1 3 5 7 9 11 13 15 17 19
CL
Example : x chart
30
40
50
CL
30
0
10
20
30
1 3 5 7 9 11 13 15 17 19
Other Control Chart
� Cumulative Sum Control Chart� Moving Average Control Chart� Geometric Moving Average Control
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Geometric Moving Average Control Chart
� Trend Chart ( Regression Control Chart )� Modified Control Chart� Acceptance Control Chart
Multivariate Control Chart
� For two or more variable simultaneously� For two variable � Hotelling’s T2 Control
Chart
32
Kata Inspirasi Hari Ini
Semakin luas Anda mengait-ngaitkan berbagai hal , semakin
33
ngaitkan berbagai hal , semakin banyak anda belajar
QuizQuiz
1. Definisikan variabel menurut anda dan contohnya ?Apa yang dapat dimonitor oleh
34
2. Apa yang dapat dimonitor oleh peta kendali variabel ?
3. Apa perbedaan fungsi X-chart dan R-chart ?