Lecture 4Lecture 4Control Chart for VariablesControl Chart for Variables

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�� 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

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• 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

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� 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

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� 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

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� 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

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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

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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

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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 ,, ==== ∑ =

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� 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 ===

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� 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

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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

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Kata Inspirasi Hari Ini

Semakin luas Anda mengait-ngaitkan berbagai hal , semakin

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ngaitkan berbagai hal , semakin banyak anda belajar

QuizQuiz

1. Definisikan variabel menurut anda dan contohnya ?Apa yang dapat dimonitor oleh

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2. Apa yang dapat dimonitor oleh peta kendali variabel ?

3. Apa perbedaan fungsi X-chart dan R-chart ?