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38
1 Six Sigma Green Belt -6 -4 -2 0 2 4 6 Introduction to Control Charts Introduction to Control Charts Sigma Quality Management

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Page 1: 17 Control Charts

1

Six SigmaGreen Belt

-6 -4 -2 0 2 4 6

Introduction to Control ChartsIntroduction to Control Charts

Sigma Quality Management

Page 2: 17 Control Charts

2

Six SigmaGreen BeltObjectivesObjectives

Be able to identify the elements of a control chart Be able to select the “best” control chart for a given indicator Understand the “theory” of how a control chart works (and why) Be able to identify and apply a rational subgrouping strategy for a

control chart

Page 3: 17 Control Charts

3

Six SigmaGreen BeltWalter ShewhartWalter Shewhart

Our Hero!

Page 4: 17 Control Charts

4

Six SigmaGreen BeltTypical Control ChartTypical Control Chart

1 3 5 7 9 11 13 15 17 19

1 3 5 7 9 11 13 15 17 19

Std. Dev.

Average

UCL

CL

UCL

LCL

CL

Subgroup

X-BAR, S CONTROL CHART

Page 5: 17 Control Charts

5

Six SigmaGreen BeltChoosing the “Best” Control ChartChoosing the “Best” Control Chart

Type of Data – Measurement vs. Count

Sample (or Subgroup) Size

Count Data Issues – Defectives vs. Defects

Page 6: 17 Control Charts

6

Six SigmaGreen BeltControl Chart SelectionControl Chart Selection

CONTROL CHART SELECTION GUIDE

What Data isto be Charted?

What type of datais to be charted?(measurement orcount)

Is a standard appliedto the entire item, or to the item's elements?

Are the count dataassumptions met?

How is the data to be

Control Chart

Questions for Count Data

DATA

Measurement

Count

Defectives

Defects

np and p chartassumptions

met

np and p chartassumptions

not met

c and u chartassumptions

met

c and u chartassumptions

not met

Subgroup size> 10

Subgroup size= 1

Subgroup sizeConstant

Subgroup sizeVarying

Constantarea of

opportunity

Varyingarea of

opportunity

X-bar, S

X, mR

np

p

X, mR

c

u

X, mR

X-bar, RSubgroup size

2 - 10

collected?

Page 7: 17 Control Charts

7

Six SigmaGreen BeltSubgroup StrategiesSubgroup Strategies

Rational Subgroup Defined

Impact of Subgrouping on Control Chart Sensitivity

Within-Group Variation

Between-Group Variation

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Mean

Standard Deviations

Total Process Variation

Time

Page 8: 17 Control Charts

8

Six SigmaGreen Belt““Simple” Explanation of Control ChartsSimple” Explanation of Control Charts

Problem of Variation – Chance vs. Assignable Causes

Criterion I – GeneralGiven a set of n data to determine whether or not they arise from a constant cause system, do the following:1. Divide the n data into m rational subgroups (of constant or variable size).2. Pick the statistics you will use to judge the data. The mean, standard deviation and proportion defective have been shown to be the most useful statistics for this purpose.3. For each statistic, calculate (using the data) estimates of the average and standard deviation of the statistic, where these estimates satisfy as nearly as possible the following conditions:

a) If the quality characteristic from which the sample is drawn is controlled with average X-Bar and standard deviation , the estimates used should approach these values as the number of data n becomes very large (i.e. in the statistical limit),

b) If the quality characteristic is not controlled, the estimates actually used should be those that will be most likely to indicate the presence of trouble (i.e. assignable causes).

4. For each statistic, construct control charts with limits based on the statistic’s estimated average plus/minus three times the statistic’s estimated standard deviation. 5. If a point falls outside the limits of the control chart, take this as evidence of the presence of assignable causes, or lack of control.

Page 9: 17 Control Charts

9

Six SigmaGreen BeltCriteria CommentsCriteria Comments

Statistics vs. Parameters� “. . One Unique Distribution . . .”� Finite Nature of Production Process� Sequence Order of the Data

Rational Subgroups

Choice of “Three Sigma”

Detecting Assignable Causes

Economy not Probability!

Page 10: 17 Control Charts

10

Six SigmaGreen BeltExercisesExercises

For your process, discuss possible subgrouping strategies - present why these could/would be “rational.”

(Optional) If you are already familiar with control charts, compare the basis for control charts presented here with your previous training.

Page 11: 17 Control Charts

11

Six SigmaGreen Belt

-6 -4 -2 0 2 4 6

Measurement Control ChartsMeasurement Control Charts

Page 12: 17 Control Charts

12

Six SigmaGreen BeltObjectivesObjectives

Be able to construct and interpret (by hand and via Minitab):� X-bar, R control charts� X, mR control charts

Page 13: 17 Control Charts

13

Six SigmaGreen BeltX-Bar, R Control ChartX-Bar, R Control Chart

1 3 5 7 9 11 13 15 17 19

Range

Average

UCL - Xbar

CL - Xbar

UCL - R

LCL - Xbar

CL - R

Subgroup

Page 14: 17 Control Charts

14

Six SigmaGreen BeltX-Bar, R Control ChartX-Bar, R Control Chart

Quality Characteristic

Before After

Changing Center

QualityCharacteristic

Before

After

QualityCharacteristic

Changing Variability

Page 15: 17 Control Charts

15

Six SigmaGreen BeltSkewed DataSkewed Data

QualityCharacteristic

Mean

QualityCharacteristic

Histogram of Averages,Samples of Size 15 Each

Page 16: 17 Control Charts

16

Six SigmaGreen BeltX-Bar. R ConstructionX-Bar. R Construction

Collect the Data – Subgroups & Size

R – Chart� Calculating Ranges

� Calculating Average Range

� Calculating Control Limits

R X X

where

X

X

R

j j j

j

j

j

max min

max

min

:

- the largest value of the " jth" subgroup

- the smallest value of the " jth" subgroup

- " jth" subgroup Range

Rk

R

R

k

R

jj

k

j

1

1

where:

- " jth" Subgroup Range

- number of Subgroups

- Average Range

UCL R D

LCL R D

where

D D

UCL

LCL

R

R

R

R

4

3

4 3

:

, - Coefficients

- Upper Control Limit for Range Chart

- Lower Control Limit for Range Chart

Page 17: 17 Control Charts

17

Six SigmaGreen BeltX-Bar, R ConstructionX-Bar, R Construction

X-Bar Chart� Calculating Subgroup Averages

� Calculating Grand Average

� Calculating Control Limits

� Drawing the Chart

xn

x

n

x

x

jj

iji

n

j

ij

j

j1

1

where:

- " jth" subgroup size

- "ith" element of the " jth" subgroup

- " jth"subgroup average

xk

x

x

jj

k

11

where:

- Grand Average of Subgroups

UCL X R A

LCL X R A

A

UCL

LCL

X Bar

X Bar

X Bar

X Bar

( )

( )

2

2

2

where:

- Coefficient

- Upper Control Limit for X - Bar

- Lower Control Limit for X - Bar

Page 18: 17 Control Charts

18

Six SigmaGreen BeltControl Chart ConstantsControl Chart Constants

Sample Size (1)

A2 D3 (2) D4 d2

2 1.880 - 3.268 1.128 3 1.023 - 2.574 1.693 4 0.729 - 2.282 2.059 5 0.577 - 2.114 2.326 6 0.483 - 2.004 2.534 7 0.419 0.076 1.924 2.704 8 0.373 0.136 1.864 2.847 9 0.337 0.184 1.816 2.970

10 0.308 0.223 1.777 3.078

Page 19: 17 Control Charts

19

Six SigmaGreen BeltX-Bar, R Control ChartX-Bar, R Control Chart

1 3 5 7 9 11 13 15 17 19

Range

Average

UCL - Xbar

CL - Xbar

UCL - R

LCL - Xbar

CL - R

Subgroup

Page 20: 17 Control Charts

20

Six SigmaGreen BeltAssignable Cause - InterpretationAssignable Cause - Interpretation

1 3 5 7 9 11 13 15 17 19

1 3 5 7 9 11 13 15 17 19

CL

1 3 5 7 9 11 13 15 17 19

CL

Rule 1:

Rule 2:

Rule 3:

Page 21: 17 Control Charts

21

Six SigmaGreen BeltAssignable Cause - InterpretationAssignable Cause - Interpretation

1 3 5 7 9 11 13 15 17 19

1 3 5 7 9 11 13 15 17 19

1

Zone

2

3

1

23

1 3 5 7 9 11 13 15 17 19

1

Zone

2

3

1

23

Rule 4:

Rule 5:

Rule 6:

Page 22: 17 Control Charts

22

Six SigmaGreen BeltAssignable Cause - InterpretationAssignable Cause - Interpretation

1 3 5 7 9 11 13 15 17 19

1

Zone

2 3

1 2 3

1 3 5 7 9 11 13 15 17 19

1

Zone

2 3

1 2 3

1 3 5 7 9 11 13 15 17 19 LCL

CL

UCL

Rule 7:

Rule 8:

Rule 9:

Page 23: 17 Control Charts

23

Six SigmaGreen BeltX, mR ConstructionX, mR Construction

Collect the Data – Subgroups & Size

R – Chart� Calculating Ranges

� Calculating Average Range

� Calculating Control Limits

� Drawing the Chart

R x x

R x x

R x x

etc

x x x x

Ri

2 2 1

3 3 2

4 4 3

2 1 2 1

.

where:

- Absolute Value of

- " ith" Subgroup Range

Rk

R

R

k

ii

k

1

1 2

where:

- Average Range

- Number of SubgroupsUCL R 3 268

3 268 4

.

( . " is the "D coefficient for the X,mR Chart)

Page 24: 17 Control Charts

24

Six SigmaGreen BeltX, mR ConstructionX, mR Construction

X Chart� Calculating Average

� Calculating Control Limits

� Drawing the Chart

Xk

X

X X

k

ii

k

i

1

1

where:

- Average of ' s

- Number of Subgroups

UCL X R

LCL X R

UCL

LCL

x

x

X

X

2 66

2 66

.

.

where:

- Upper Control Limit for X

- Lower Control Limit for X

Page 25: 17 Control Charts

25

Six SigmaGreen BeltX, mR Control ChartX, mR Control Chart

1 3 5 7 9 11 13 15 17 19

Range

Individuals

UCL - X

CL - X

UCL - R

LCL - X

CL - R

Subgroup

Page 26: 17 Control Charts

26

Six SigmaGreen Belt

-6 -4 -2 0 2 4 6

Attribute Control ChartsAttribute Control Charts

Page 27: 17 Control Charts

27

Six SigmaGreen BeltObjectivesObjectives

Be able to construct and interpret (by hand and Minitab):� P & np control charts� C & u control charts

Page 28: 17 Control Charts

28

Six SigmaGreen BeltAttribute Control ChartsAttribute Control Charts

‘Defective” Defined

“Defects” Defined

Binomial Assumptions – np & p Control Charts

Poisson Assumptions – c & u Control Charts (later)

Page 29: 17 Control Charts

29

Six SigmaGreen BeltAssignable Causes – Attribute ChartsAssignable Causes – Attribute Charts

1 3 5 7 9 11 13 15 17 19

1 3 5 7 9 11 13 15 17 19

CL

1 3 5 7 9 11 13 15 17 19

CL

Rule 1:

Rule 2:

Rule 3:

1 3 5 7 9 11 13 15 17 19

Rule 4:

Page 30: 17 Control Charts

30

Six SigmaGreen BeltnP Control ChartnP Control Chart

Collecting the Data

Counting the Number of Defectives

Calculating Average No. of Defectives

Calculating UCL, LCL

Drawing the Chart

subgroupper Defectives ofNumber Average -

Subgroups ofNumber -

subgroup ith"" Items, Defective ofNumber -

:where1

pn

k

np

knppn

i

k

ii

Subgroups ofNumber -

Size Subgroup ith"" -

:where

1

1

k

n

nk

n

i

k

ii

UCL np np np n

LCL np np np n

UCL

LCL

n

np

np

np

np

3 1

3 1

( )

( )

where:

- Upper Control Limit

- Lower Control Limit

- Constant (or Average) Subgroup Size

Page 31: 17 Control Charts

31

Six SigmaGreen BeltnP Control ChartnP Control Chart

1 3 5 7 9 11 13 15 17 19

# Defective

CL

UCL

LCL

Page 32: 17 Control Charts

32

Six SigmaGreen Beltp Control Chartp Control Chart

Collecting the Data

Calculating the Fraction Defectives

Calculating Average Fraction Defectives

Calculating UCL, LCL

Drawing the Chart

subgroup ith"" - defectiveFraction -

subgroup ith"" - size Subgroup -

subgroup ith""- defectiveNumber -

:where

%100

:Defective)Percent (or DefectiveFraction

i

i

i

i

ii

i

ii

p

n

np

n

npp

n

npp

%100 = defectivepercent or,

defectivefraction Average -

:where11

p

p

nnppk

ii

k

ii

size subgroup ith"" -

Limits ControlLower & Upper - LC,

:where

/)100(3 and /)100(3

:defectivepercent for or,

/)1(3 and /)1(3

i

pp

ipip

ipip

n

LUCL

npppLCLnpppUCL

npppLCLnpppUCL

Page 33: 17 Control Charts

33

Six SigmaGreen Beltp Control Chartp Control Chart

1 3 5 7 9 11 13 15 17 19

% Defective

CL

Assignable Cause

Subgroup

Page 34: 17 Control Charts

34

Six SigmaGreen Beltc & u Control Chartsc & u Control Charts

Poisson Assumptions for c & u Charts

Page 35: 17 Control Charts

35

Six SigmaGreen Beltc Control Chartc Control Chart

Collecting the Data

Counting the Number of Defects

Calculating Average No. of Defects

Calculating UCL, LCL

Drawing the Chart

ck

c

c

k

c

ii

k

i

1

1

where:

- Number of defects, "ith" subgroup

- Number of subgroups

- Average number of defects

UCL c c

LCL c c

UCL

LCL

c

c

c

c

3

3

where:

- Upper Control Limit

- Lower Control Limit

Page 36: 17 Control Charts

36

Six SigmaGreen Beltc Control Chartc Control Chart

1 3 5 7 9 11 13 15 17 19

# Defects

CL

UCL

LCL

Page 37: 17 Control Charts

37

Six SigmaGreen Beltu Control Chartu Control Chart

Collecting the Data

Counting the Number of Defects & Defect Rate/Subgroup

Calculating Average Rate of Defects

Calculating UCL, LCL

Drawing the Chart

subgroup ith"" rate,Defect -

subgroupith"" y,opportunit of Area -

subgroup ith"" defects, ofNumber -

:where

i

i

i

i

ii

u

n

c

n

cu

subgroups ofNumber -

defects ofnumber Average -

:where11

k

u

ncuk

ii

k

ii

Limit ControlLower -

Limit Control Upper -

:where

/3

/3

u

u

iu

iu

LCL

UCL

nuuLCL

nuuUCL

Page 38: 17 Control Charts

38

Six SigmaGreen Beltu Control Chartu Control Chart

1 3 5 7 9 11 13 15 17 19

Defect Rate

CL

Assignable Cause

Subgroup