1

NİCELİKSEL KONTROL

GRAFİKLERİ

Prof. Dr. Nihal ERGİNEL

Anadolu Üniversitesi

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X BİRİMLER VE

HAREKETLİ DEĞİŞİM ARALIĞI

KONTROL GRAFİĞİ

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X- Birimler Kontrol Grafiği

n= 1 birimlik örnekler alınır.

Üretim hızı oldukça düşük, üretim sayısı az

olduğu durumlarda,

Otomatik ölçüm cihazları ile her birimin

ölçümü yapılabildiği zamanlarda,

Üretim sürecinde değişkenlik çok az

olduğu durumlarda,

Test metodu tahribatlı olduğunda,

tercih edilir.

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2

X birimler kontrol grafiği, Hareketli Değişim Aralığı Kontrol Grafiği ile birlikte kullanılır.

Hareketli Değişim Aralığı KG, birbirini izleyen iki veri arasındaki değişkenliği gösterir.

Stat>Control Charts>Variables Charts for Individuals

>I-MR

Stat>Control Charts>Variables Charts for Individuals>

Individuals

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Eğer ana kütle µ ve σ biliniyor ise;

ÜKS= 0 + 3

OÇ = 0

AKS= 0 – 3

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Veri toplanması ve Analizi

n=3 olduğu durumda;

t=1 = max ( , , ) - min ( , , )

t=2 = max ( , , ) - min ( , , )

. . .

. . .

. .

. .

t=m = max ( , , ) - min ( , , )

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3

n=3 için;

=

=

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X- Birimler Kontrol Grafiğinin sınırlarının

hesaplanması

m: örnek sayısı

n: hareketli değişim aralığı

= + 3

=

= =

= - 3

=

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Hareketli Ortalama (MR) Kontrol Grafiği

sınırlarının hesaplanması

= .

=

= .

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4

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X-HDA Kontrol Grafiği

2010Subgroup 0

4100

4000

3900

3800

3700

3600

3500

Indiv

idual V

alu

e

1

Mean=3789

UCL=4026

LCL=3553

300

200

100

0

Movin

g R

ange

R=88,89

UCL=290,4

LCL=0

I and MR Chart f or C1

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L değeri için X-HDA KG

40302010Subgroup 0

81

80

79

78

77

Indiv

idual V

alu

e

1

1

1

1

1

1

1 1

1 1 1 1 1

Mean=79,87

UCL=80,81

LCL=78,92

1,0

0,5

0,0

Movin

g R

ange

1

R=0,356

UCL=1,163

LCL=0

I and MR Chart f or L-pembe

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5

a değeri için X-KG

403020100

8

7

6

5

Observation Number

Indiv

idual V

alu

e

I Chart f or a-pembe

1

11

1

11

1 1

Mean=5,833

UCL=6,940

LCL=4,726

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b değeri için X- KG

403020100

5

4

3

Observation Number

Indiv

idual V

alu

e

I Chart f or b-pembe

11 1

1

Mean=3,828

UCL=4,563

LCL=3,093

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ÖRNEK:

Uçak astar boyası için viskozite değeri önemli kalite

karakteristiklerinden biridir. Ürünler partiler halinde

üretilmektedir ve 15 partiye ait viskozite değeri aşağıdaki

tabloda verilmiştir. Verileri dikkate alarak X birimler ve

MR kontrol grafiğini çiziniz.

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6

Parti

numarası

Viskozite

1 33,75

2 33,05

3 34,00

4 33,81

5 33,46

6 34,02

7 33,68

8 33,27

9 33,49

10 33,20

11 33,62

12 33,00

13 33,54

14 33,12

15 33,84

Çözüm:

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Çözüm: Parti

numarası

Viskozite

(X)

MR

1 33,75

2 33,05 0,7

3 34,00 0,95

4 33,81 0,19

5 33,46 0,35

6 34,02 0,56

7 33,68 0,34

8 33,27 0,41

9 33,49 0,22

10 33,20 0,29

11 33,62 0,42

12 33,00 0,62

13 33,54 0,54

14 33,12 0,42

15 33,84 0,72

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X- Birimler Kontrol Grafiği Sınırları;

n= 2 olduğu için =1,128 olur.

= + 3

= 33,52+ 3

= 34,80

= =33,52

= - 3

= 33,52- 3

= 32,24

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7

X birimler kontrol grafiği

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Hareketli Değişim Aralığı (MR) Kontrol

Grafiği Sınırları;

n= 2 için = 0 ve = 3,267 olur.

= . = 3,267.0,48=1,57

= =0,48

= . = 0

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8

– R KONTROL GRAFİĞİ

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X-R Kontrol Grafiği

Süreçteki anlık ve belli bir periyottaki

değişimleri görmek için kullanılır,

Süreçten n (3,5,vb.) birimlik örnekler alınır,

En yaygın olarak kullanılan kontrol

grafiğidir.

zamann birimlik

örnekn birimlik

örnek

n birimlik

örnek...

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X ortalama, n birimlik örneklerin ortalamasını, R ise n

birimlik örneklerin max. ile min. arasındaki farkı gösterir.

Stat>Control Charts>Variables Charts for Subgroups

> Xbar-R

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9

Eğer ana kütle µ ve σ biliniyor ise;

kontrol grafiği;

= +3

= + Aσ

= =

= -3

= – Aσ

A =

alındığında;

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R kontrol grafiği;

= .σ

= .σ

= .σ

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Ancak her zaman ana kütle parametreleri

bilinmez. Böyle durumlarda, ana kütle

standart sapmasını ya örnek standart

sapmasından veya değişim aralığından

tahmin etmek gerekir.

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10

Veri toplama

t = 1

t =2

t =3

.

.

.

t= m

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=

=

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X-R Kontrol Grafiği sınırlarının

hesaplanması

X: ölçüm değeri

n: örnekteki birim sayısı

örnek ortalaması

: genel ortalama

R: örnek değişim aralığı

değişim aralığı ortalaması

, , : sabitler

olmak üzere,

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11

kontrol grafiği

= + 3

= + .

=

= - 3

= - .

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R kontrol grafiği;

= .

=

= .

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Veri girme

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12

Kontrol Grafiğini oluşturma

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L değeri için X- R KG

0Subgroup 10 20 30

78

79

80

81

82

Sam

ple

Mean

Mean=80,42

UCL=81,03

LCL=79,81

0,0

0,5

1,0

1,5

Sam

ple

Range

1

R=0,2294

UCL=0,7494

LCL=0

Xbar/R Chart f or L-Gri

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a değeri için X- R KG

0Subgroup 10 20 30

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

Sam

ple

Mean

Mean=0,3665

UCL=0,6163

LCL=0,1168

0,0

0,1

0,2

0,3

0,4

Sam

ple

Range

R=0,09392

UCL=0,3069

LCL=0

Xbar/R Chart f or a-Gri

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b değeri için X- R KG

0Subgroup 10 20 30

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Sam

ple

Mean

Mean=0,1909

UCL=0,3293

LCL=0,05256

0,0

0,1

0,2

Sam

ple

Range

R=0,05202

UCL=0,1700

LCL=0

Xbar/R Chart f or b-Gri

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ÖRNEK:

Otomobil motorundaki piston çaplarının kontrol altında

olup olmadığının belirlenmesi için süreçten 5 birimlik

toplam 25 örnekler alınmıştır. İlgili analizleri yaparak

yorumlayınız.

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GÖZLEMLER

74,03 74,002 74,019 73,992 74,008

73,995 73,992 74,001 74,011 74,004

73,988 74,024 74,021 74,005 74,002

74,002 73,996 73,993 74,015 74,009

73,992 74,007 74,015 73,989 74,014

74,009 73,994 73,997 73,985 73,993

73,995 74,006 73,994 74 74,005

73,985 74,003 73,993 74,015 73,998

74,008 73,995 74,009 74,005 74,004

73,998 74 73,998 74,007 73,995

73,994 73,998 73,994 73,995 73,99

74,004 74 74,007 74 73,996

73,983 74,002 73,998 73,997 74,012

74,006 73,967 73,994 74 73,984

74,012 74,014 73,998 73,999 74,007

74 73,984 74,005 73,998 73,996

73,984 74,012 73,986 74,005 74,007

74,006 74,01 74,018 74,003 74

73,984 74,002 74,003 74,005 73,997

74 74,01 74,013 74,02 74,003

73,988 74,001 74,009 74,005 73,996

74,004 73,999 73,99 74,006 74,009

74,01 73,989 73,99 74,009 74,014

74,015 74,008 73,993 74 74,01

73,982 73,984 73,995 74,017 74,013

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14

Çözüm:GÖZLEMLER Ortalama R

74,03 74,002 74,019 73,992 74,008 74,01 0,038

73,995 73,992 74,001 74,011 74,004 74,006 0,019

73,988 74,024 74,021 74,005 74,002 74,008 0,036

74,002 73,996 73,993 74,015 74,009 74,003 0,022

73,992 74,007 74,015 73,989 74,014 74,003 0,026

74,009 73,994 73,997 73,985 73,993 73,996 0,024

73,995 74,006 73,994 74 74,005 74 0,012

73,985 74,003 73,993 74,015 73,998 73,997 0,03

74,008 73,995 74,009 74,005 74,004 74,004 0,014

73,998 74 73,998 74,007 73,995 73,998 0,017

73,994 73,998 73,994 73,995 73,99 73,994 0,008

74,004 74 74,007 74 73,996 74,001 0,011

73,983 74,002 73,998 73,997 74,012 73,998 0,029

74,006 73,967 73,94 74 73,984 73,99 0,039

74,012 74,014 73,998 73,999 74,007 74,006 0,016

74 73,984 74,005 73,998 73,996 73,997 0,021

73,984 74,012 73,986 74,005 74,007 74,001 0,026

74,006 74,01 74,018 74,003 74 74,007 0,018

73,984 74,002 74,003 74,005 73,997 73,998 0,021

74 74,01 74,013 74,02 74,003 74,009 0,02

73,988 74,001 74,009 74,005 73,996 74 0,033

74,004 73,999 73,99 74,006 74,009 74,002 0,019

74,01 73,989 73,99 74,009 74,014 74,002 0,025

74,015 74,008 73,993 74 74,01 74,005 0,022

73,982 73,984 73,995 74,017 74,013 73,998 0,035

Ort.: 74,001 ort:0,023

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Kontrol Grafiği Sınırlarının Hesaplanması:

=

=

=

= 74,001

UCL = = + . =74,001 + (0,577) (0,023)= 74,014

LCL = = - . =74,001 - (0,577) (0,023)= 73,98

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252321191715131197531

74,020

74,015

74,010

74,005

74,000

73,995

73,990

73,985

Sample

Sa

mp

le M

ea

n

__X=74,00112

UCL=74,01730

LCL=73,98494

Xbar Chart of C1

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Kontrol Grafiği Sınırlarının Hesaplanması:

=

=

=

=0,023

n = 5 için =0 ve = 2,115

UCL = . =2,115.0,023=0,049

LCL= . = 0.0,023=0

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Kontrol Grafiği

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– S KONTROL GRAFİĞİ

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16

– S KONTROL GRAFİĞİ

n >1 olduğu zaman değişkenliği belirleme

için S (örnek standart sapması) grafiği

kullanılır.

S =

Stat>Control Charts>Variables Charts for

Subgroups> X Bar-S

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S =

=

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kontrol grafiği sınırları

= +3

= +

=

= -3

= -

=

=

olarak tanımlanır ise,

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17

S kontrol grafiği sınırları

S kontrol grafiği;

=

=

=

= Standart sapma S’nin standart sapması

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= . + 3

= .

= . - 3

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biliniyor ise;S-Kontrol Grafiği

= .

= .

= .

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18

Örnek:GÖZLEMLER Ortalama R

74,03 74,002 74,019 73,992 74,008 74,01 0,038

73,995 73,992 74,001 74,011 74,004 74,006 0,019

73,988 74,024 74,021 74,005 74,002 74,008 0,036

74,002 73,996 73,993 74,015 74,009 74,003 0,022

73,992 74,007 74,015 73,989 74,014 74,003 0,026

74,009 73,994 73,997 73,985 73,993 73,996 0,024

73,995 74,006 73,994 74 74,005 74 0,012

73,985 74,003 73,993 74,015 73,998 73,997 0,03

74,008 73,995 74,009 74,005 74,004 74,004 0,014

73,998 74 73,998 74,007 73,995 73,998 0,017

73,994 73,998 73,994 73,995 73,99 73,994 0,008

74,004 74 74,007 74 73,996 74,001 0,011

73,983 74,002 73,998 73,997 74,012 73,998 0,029

74,006 73,967 73,94 74 73,984 73,99 0,039

74,012 74,014 73,998 73,999 74,007 74,006 0,016

74 73,984 74,005 73,998 73,996 73,997 0,021

73,984 74,012 73,986 74,005 74,007 74,001 0,026

74,006 74,01 74,018 74,003 74 74,007 0,018

73,984 74,002 74,003 74,005 73,997 73,998 0,021

74 74,01 74,013 74,02 74,003 74,009 0,02

73,988 74,001 74,009 74,005 73,996 74 0,033

74,004 73,999 73,99 74,006 74,009 74,002 0,019

74,01 73,989 73,99 74,009 74,014 74,002 0,025

74,015 74,008 73,993 74 74,01 74,005 0,022

73,982 73,984 73,995 74,017 74,013 73,998 0,035

Ort.: 74,001 ort:0,023

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X-ortalama Kontrol Grafiği

UCL = = + . =74,001 + (1,4277) (0,0094)= 74,014

LCL = = - . =74,001 - (1,4277) (0,0094)= 73,98

=

=

(1850,028) = 74,001

=

=

(0,2351)=0,0094

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S Kontrol Grafiği

UCL = . = (2,089) (0,0094)= 0,0196

CL= 0,0094

LCL = . =0.(0,0094)= 0

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19

X-ortalama Kontrol Grafiği

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S Kontrol Grafiği

Her iki kontrol grafiğine göre

süreç kontrol altındadır.

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5 a et örnek i in 0 9400 tür ürecin stan art sapması ise;

=

=

0 0

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