Module-7Control Charts

2

Control Charts – Learning Objectives

At the end of this section, delegates will:

• Understand how control charts can show if a

process is stable

• Generate and interpret control charts for variable

and attribute data

• Understand the role of control charts within the

DMAIC improvement process

3

Control Charts – Agenda1. Introduction to Statistical Process Control, SPC

2. Control Limits

3. Individual and Moving Range Chart

4. Workshop on Control Charts

5. Defective (Binomial) p-Chart

6. Defects (Poisson) u-Chart

7. Workshop on Attribute Control Charts

8. Uses of Control Charts

9. Summary

4

What is Statistical Process Control?

• Statistical Process Control is a method of monitoring and detecting changes in processes.

• SPC uses an advanced form of Time Series plots.

• SPC provides an easy method of deciding if a process has changed (in other words, is the process “in-control”?).

5

We Need Ways of Interpreting Data

• Everyday we are flooded by data and we are

forced to make decisions:

• Calls handled decreases by 4%

• UK trade deficit rises by £5 billion

• Company X’s earnings are $240Million less

than the previous quarter

• Should we take action ?

6

Leave it alone -it ain’t broke

Pain &suffering

Pain &suffering

Lower “Customer”Requirement

Upper “Customer”Requirement

How do we manage data historically?

This Method

• Tells you where you are in relation to customer’s needs

• It will NOT tell you how you got there or what to do next

• Means that pressure to achieve customer requirements will cause you to:

• Actually Fix The Process

• Sabotage The Process

• Sabotage The Data (Integrity)

7

What do Control Charts detect?

• Control Charts detect changes in a process.

• All processes change slightly, but process control

aims to detect ‘statistically significant’ changes that

are not just random variation.

• Processes can change in several different ways…

• the process average can change

• the process variation can change

• the process may contain one-off events

8

Process Control

• Process control refers to the evaluation of process stability over time

• Process Capability refers to the evaluation of how well a process meets specifications

LSL USL

0 5 10 15 20 25

Time

UCL

LCL

9

Why would a Process be Incapable?There are a number of reasons why a process may not be capable of meeting specification:

1. The specification is incorrect!

2. Excessive variation

3. The process is not on target

4. A combination of the above

5. Errors are being made

6. The process is not stable

10

The specification is incorrect

• This issue was discussed during the Customer Focus section of this course

• If specifications are not clearly related to customer requirements, then it is always a good idea to challenge the specification before attempting to improve the process

11

Excessive variationUpper

Specification

Limit

Lower

Specification

LimitTarget

• Excessive variation means that we have a variation reduction issue

• We will need to understand which process inputs are causing the variation in the process output

12

The process is not on targetUpper

Specification

Limit

Lower

Specification

LimitTarget

• In this situation we have a process targeting issue

• We will need to understand which process inputs are causing the process to be off-target

• This situation is sometimes simple to solve!

13

Excessive variation and not on targetUpper

Specification

Limit

Lower

Specification

LimitTarget

• In this situation we have both excessive variation and a process targeting issue

• We will need to understand which process inputs are causing the excessive variation and which are causing the process to be off-target

14

Errors are being madeUpper

Specification

Limit

Lower

Specification

LimitTarget

• A situation such as this might indicate that errors are being made which result in occasional excursions outside of the specification

• This is often an indication of a mistake proofing issue

15

The process is not stable

• A situation such as this is an indication that the process is unstable

• Whenever this situation is encountered in a DMAIC activity, then the reason(s) for the instability must be found and removed before assessing process capability

Upper

Specification

Limit

Lower

Specification

Limit Target

Last week

This week

Next week?

16

The process is not stableUpper

Specification

Limit

Lower

Specification

LimitTarget

Last week

This week

Next

week?

• Causes of process instability are sometimes referred

to as “special causes”

• Removing these special causes may result in the

process becoming capable of consistently meeting the

target

17

Unstable Process

“Special”causes of variation are present

Tim

e

TotalVariation

Target

18

Stable Process

Tim

eTarget

TotalVariation

Only “Common”causes of variation are present

19

Capable Process

Tim

e

Spec Limits

CAPABLE

NOT

CAPABLE Process is Stable but Process is not Capable

Process is Stable and Process is Capable

Management Action

(DMAIC) to reducecommon cause variation

20

Control Charts test for Stability(Control Chart of Average)

0 5 10 15 20 25

Time

1.26

1.27

1.28

1.29

1.3

Pro

cess A

vera

ge

Upper Control Limit

Lower Control Limit

21

Transactional Improvement Process

� Select Project

� Define Project

Objective

� Form the Team

� Map the Process

� Identify Customer Requirements

� Identify Priorities

� Update Project File

� Select Project

� Define Project

Objective

� Form the Team

� Map the Process

� Identify Customer Requirements

� Identify Priorities

� Update Project File

� Control Critical x ’s

� Monitor y’s

� Validate Control Plan

� Identify further opportunities

� Close Project

15 20 25 30 35

LSL USL

Phase Review

1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

y

Phase Review

� Develop Detailed Process Maps

� Identify Critical Process Steps (x ’s) by looking for:

– Process Bottlenecks

– Rework / Repetition

– Non-value Added Steps

– Sources of Error / Mistake

� Map the Ideal Process

� Identify gaps between current and ideal

START

PROCESSSTEPS

DECISION

STOP

Phase Review

� Brainstorm Potential Improvement Strategies

� Select Improvement Strategy

� Plan and Implement Pilot

� Verify Improvement

� Implement Countermeasures

Criteria A B C D

Time + s - +

Cost + - + s

Service - + - +

Etc s s - +

15 20 25 30 35

LSL USL

Phase Review

Analyse Improve ControlMeasureDefine

Phase Review

Define

� Define Measures (y ’s)

� Check Data Integrity

� Determine Process Stability

� Determine Process Capability

� Set Targets for Measures

15 20 25 30 35

LSL USL

15 20 25 30 35

LSL USL

Phase Review

1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

y

Phase Review

� Develop Detailed Process Maps

� Identify Critical Process Steps (x ’s) by looking for:

– Process Bottlenecks

– Rework / Repetition

– Non-value Added Steps

– Sources of Error / Mistake

� Map the Ideal Process

� Identify gaps between current and ideal

START

PROCESSSTEPS

DECISION

STOP

Phase Review

� Brainstorm Potential Improvement Strategies

� Select Improvement Strategy

� Plan and Implement Pilot

� Verify Improvement

� Implement Countermeasures

Criteria A B C D

Time + s - +

Cost + - + s

Service - + - +

Etc s s - +

15 20 25 30 35

LSL USL

Phase Review

Analyse Improve ControlMeasureDefine

Phase Review

Define

� Define Measures (y ’s)

� Check Data Integrity

� Determine Process Stability

� Determine Process Capability

� Set Targets for Measures

22

Role of Control ChartsMeasure Phase:

• used during capability studies to assess process stability

Improve Phase:

• used to establish if the modified, improved process is stable

Control Phase

• used to control critical process input variables (x’s) in order to

reduce variability in process outputs (y’s)

• used to monitor process outputs (y’s) on an ongoing basis to

ensure that the process remains in control

23

Control Charts

VariableData

NoSubgroups

Subgroupsn = 2-9

Subgroupsn > 9

Individuals &Moving Range

Chart

X Bar & RChart

X Bar & sChart

AttributeData

Defect Data(Poisson)

Defective Data(Binomial)

VaryingSubgroup

Size

ConstantSubgroup

Size

VaryingSubgroup

Size

ConstantSubgroup

Size

u Chart c Chart p Chart np Chart

24

What do Control Charts tell us?

• Is the process stable? • Should we be taking action?• Are there any special causes?• What is the average process output?• What is the variability?

0 5 10 15 20 25

Subgroup

1.26

1.27

1.28

1.29

1.3

X-b

ar

Control Limits

26

Total Variation

TotalVariation

Within SubgroupVariation

Between SubgroupVariation

27

Control Limits Use Within Subgroup Variation

• The total variation and Within Subgroup variation are the same only if the process is stable

• The Within Subgroup variation is an estimate of what the total variation would be if the process were stable

• The Within Subgroup variation is used to calculate the control limits since these limits represent the range of values expected for a stable process

28

Controls Limits

• Controls limits are always:

Average ± 3 Standard DeviationsWhere the average and standard deviation are the

average and standard deviation of whatever data

is plotted:

99.7%99.7%99.7%99.7%

−−−−4σ4σ4σ4σ −−−−3σ3σ3σ3σ −−−−2σ2σ2σ2σ −−−−1σ1σ1σ1σ +1σ+1σ+1σ+1σ +2σ+2σ+2σ+2σ +3σ+3σ+3σ+3σ +4σ+4σ+4σ+4σ0

29

Control LimitsUpper Control Limit

Lower Control Limit

• Control Limits are statistical boundaries which tell us whether or not the process is stable

• Based on the normal distribution, 99.7% of the points plot within the control limits if the process is stable

• The chance of a point outside the control limits, falsely indicating the process is unstable, is only 0.3% or 1 in 370

Individuals Control Chart

31

Individuals Control Chart• Used when only a single observation per time period

(subgroup):� Monthly reporting data:

• On-time shipments, In-process Inventory, Complaints, etc.

� Rare events

� Sales

� Stock Price

� Inventory Levels

� Customer Response Time

� Lost Time Accidents

� Complaints

� Anything that can be measured and varies

32

Within Subgroup Variation• The best estimate is obtained by taking the differences between

consecutive samples i.e. the Moving Range (MR).

• We can use the the average MR, R, or the median MR, R

• When using R the Short Term standard deviation is estimated by:

1.128

R

d

Rσ

2

Within ==

˜

• When using R the standard deviation is estimated by:˜

0.954

R~

d

R~

σ

4

Within==

33

Table of Constants for ImR chartsSample size d 2 d 3 d 4 D 3 D 4 D 5 D 6 E 2 E 5

2

3

4

56

7

8

9

10

0.853

0.888

0.880

0.8640.848

0.833

0.820

0.808

0.797

0.954

1.588

1.978

2.2572.472

2.645

2.791

2.915

3.024

3.267

2.574

2.282

2.1142.004

1.924

1.864

1.816

1.777

2.970

3.078

3.865

2.744

2.376

0

0

0

0 2.179

0.209

1.075

1.029

0.9921.809 0.975

0

0.055

0.119

0.168

2.054

1.967

1.901

1.850

1.128

1.693

2.059

2.3262.534

2.704

2.847

0

0

0

00

0.076

0.136

0.184

0.223

2.660

1.772

1.457

1.2901.184

1.109

1.054

1.010

3.145

1.889

1.517

1.3291.214

1.134

We would generally calculate the differences between consecutive samples, which corresponds to a “sample size” of 2 in this table.Minitab will calculate the control chart limits for us!

34

Control Limits

The controls limits for the average, based on R are:

R2.66XREX1.128

R3X

d

R3X3σX 2

2

Within ±=±=±=±=±

The controls limits for the range, based on R are:

R3.267RDUCL

0R0RDLCL

4Range

3Range

==

=×==

35

Control Limits

R~

3.145XR~

EX0.953

R~

3Xd

R~

3X3σX 5

4

Within ±=±=±=±=±

~The controls limits for the average, based on R are:

The controls limits for the range, based on R are:~

R~

3.865R~

DUCL

0R~

0R~

DLCL

6Range

5Range

==

=×==

36

R Versus R• Some of the differences may be contaminated by

shifts in the mean (special Causes).

• R, the median MR, is more robust to this contamination so is generally preferred.

• When many of the differences are zero, it might be necessary to use R instead.

• A conversion factor can be developed:

182.1R~

954.0

128.1R~

R

954.0

R~

128.1

Rwithin

××××====××××====

====σσσσ====

~

¯ ˜

37

Call Out TimeSample Number Call Out Time

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

5.35

3.28

1.07

1.06

4.29

3.23

5.40

6.42

3.25

8.55

4.26

7.48

5.35

2.14

4.24

6.44

3.21

9.66

4.28

5.33

38

Call Out TimeSample Number Call out Time

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Difference

5.35

3.28

1.07

1.06

4.29

3.23

5.40

6.42

3.25

8.55

4.26

7.48

5.35

2.14

4.24

6.44

3.21

9.66

4.28

5.33

2.07

2.21

0.01

3.23

1.06

2.07

1.02

5.30

4.29

3.22

2.13

3.21

2.10

2.20

3.23

6.45

5.38

1.05

2.21

4.71 2.82

Average Average Difference

Ordered

0.01

1.02

1.05

1.06

2.07

2.07

2.10

2.13

2.20

2.21

2.21

3.21

3.22

3.23

3.23

4.29

5.30

5.38

6.45

Median

39

Individuals chart - MinitabOpen Worksheet: Call Out Time

Stat>Control Charts>Variables Charts for Individuals>I-MR

Select Variable: Call Out Time

Click “I-MR” Options

Click “Estimate” – select “Median moving range”

Click “Tests” - select “1 point > 3 standard deviations from center line”

40

IMR Chart – Minitab (using R)

Observation

Indiv

idual Valu

e

2018161412108642

12

8

4

0

_X=4.71

UC L=11.66

LC L=-2.24

Observation

Movin

g R

ange

2018161412108642

8

6

4

2

0

__MR=2.613

UC L=8.538

LC L=0

I-MR Chart of Call out Time

˜

613.2182.1R~

R

21.2R~

Median

====××××====

====

41

Lognormal and other non –normal data

• When dealing with lognormal and other non-normal data

we need to be cautious.

• X-bar and Range charts will be acceptable for most non-

normal data with a sub-group size of 5 or larger.

• I-MR charts may give false indications of instability.

42

Workshop – Individuals Control Chart

Open Minitab worksheet: PAYMENT TIMES.MTW

Use Minitab to create:

• Individuals and Moving Range chart

• Use the Median with a moving range of 2

• Assess the stability of the process

• What would you want to do next?

• Prepare a short report of your findings

Attribute Control Charts

44

Control Charts for Defective Items (Binomial)

• A p-Chart is used to track the proportion defective

• The p-Chart is constructed using data on the number of defectives from varying (or fixed) subgroup sizes

• The data opposite shows the number of defective orders from random samples taken over 10 working days

• The subgroups should be large enough to contain 5 or more defective items

Sample

Number

Defectives

(np)

Subgroup

Size

1 8 96

2 12 104

3 13 99

4 8 100

5 7 103

6 13 110

7 6 97

8 7 88

9 10 111

10 8 105

45

Control Charts for Defective Items (Binomial)

• The p-chart must satisfy the requirements for the Binomial

Distribution. The particular requirements affecting the p-chart are:

1. Each unit (e.g., transaction, invoice, …) can only be classified

as pass or fail

2. If one unit (e.g., transaction, invoice, …) fails, then the chance

of the next unit failing is not affected

• If the Binomial distribution is not appropriate then it may be

possible to use the Individuals control chart already discussed

• Since we are charting defective items this chart should not be

used when the number of defectives is zero or there are a large

number of zeros (80-90%)

46

P Chart Construction

Sample

Number

Defectives

(np)

Subgroup

Size (n)

Proportion

Defective

(p)

Average

Proportion

Defective

(pbar) 3 Sigma UCL(p) LCL(p)

1 8 96 0.083 0.091 0.088062 0.179062 0.002938

2 12 104 0.115 0.091 0.084608 0.175608 0.006392

3 13 99 0.131 0.091 0.086718 0.177718 0.004282

4 8 100 0.08 0.091 0.086283 0.177283 0.004717

5 7 103 0.068 0.091 0.085017 0.176017 0.005983

6 13 110 0.118 0.091 0.082268 0.173268 0.008732

7 6 97 0.062 0.091 0.087607 0.178607 0.003393

8 7 88 0.08 0.091 0.091978 0.182978 -0.00098

9 10 111 0.091 0.091 0.081896 0.172896 0.009104

10 8 105 0.076 0.091 0.084204 0.175204 0.006796

Total 92 1013

n

)p-(1p3-p3σpLCL,

n

)p-(1p3p3σpUCL

n

)p-(1pσ0.091,

1013

92

Σn

Σnpp

pp =−=+=+=

====

47

P Chart - Minitab

Open Worksheet P Chart

Stat>Control Charts>Attributes Charts>P

Variable: Defectives

Subgroups in: “Subgroup Size” Click “P Chart – Options”

Click “Tests” – select “1 point > 3 standard deviations from center line”

Sample

Pro

port

ion

10987654321

0.20

0.15

0.10

0.05

0.00

_P=0.0908

UCL=0.1749

LCL=0.0067

P Chart of Defectives

Tests performed with unequal sample sizes

48

Control Charts for Defects (Poisson)

• A u Chart is used to track the number of defects per unit (e.g., transaction, invoice, …).

• The u chart is constructed using data on the number of defects from varying subgroup sizes (number of units).

• The data opposite shows the number of defects in the given number of invoices sampled randomly from 10 weeks of invoicing.

Sample

Number

Defects

c

Invoices

n

1 7 40

2 4 45

3 8 33

4 5 40

5 3 39

6 8 46

7 5 27

8 7 45

9 9 38

10 4 39

49

Control Charts for Defects (Poisson)

• Since the u-chart is based on the Poisson Distribution, the data should be tested to see if it fits the Poisson distribution (e.g., some “count data”such as complaints and late shipments may not fit the Poisson distribution)

• If the Poisson distribution does not fit then it may be possible to use the Individuals control chart already discussed

• Since we are charting defects, this chart should not be used when the number of defects is zero or there are a large number of zeros (80-90%)

50

U Chart - ConstructionSample

Number

Defects

c

Invoices

n

DPU

u u bar LCL(u) UCL(u)

1 7 40 0.175 0.153 0 0.34

2 4 45 0.089 0.153 0 0.328

3 8 33 0.242 0.153 0 0.307

4 5 40 0.125 0.153 0 0.339

5 3 39 0.077 0.153 0 0.341

6 8 46 0.174 0.153 0 0.326

7 5 27 0.185 0.153 0 0.378

8 7 45 0.156 0.153 0 0.327

9 9 38 0.237 0.153 0 0.343

10 4 39 0.103 0.153 0 0.341

Total 60 392 0.153

n

u3u3uLCL,

n

u3u3uUCL

uσ 0.153,392

60

Σn

Σcu

uu −−−−====−−−−====++++====++++====

================

σσσσσσσσ

51

U Chart - Minitab

Open Worksheet: U Chart

Stat>Control Charts>Attributes Charts>U

Variable: Defects Subgroups in: “Units”

Click “U Chart Options”

Click “Tests” – select “1 point > 3 standard deviations from center line”

Sample

Sample C

ount Per Unit

10987654321

0.4

0.3

0.2

0.1

0.0

_U=0.1531

UCL=0.3410

LCL=0

U Chart of Defects

Tests performed with unequal sample sizes

52

Workshop - Attributes Control Chart

• Using the packets of sweets provided (assume that each packet has been taken from a different batch of production over the last few days):

� Randomly select 20 sweets from each packet

� Inspect the sweets for two types of defect-

• Badly mis-shaped/damaged sweet

• Missing or poorly printed logo

• Using Minitab, assess the stability of the process

• Prepare a short report of your findings

Uses of Control Charts

54

Control Charts

Variable

Data

No

Subgroups

Subgroups

n = 2-9

Subgroups

n > 9

Individuals &

Moving Range

Chart

X Bar & R

Chart

X Bar & s

Chart

Attribute

Data

Defect Data

(Poisson)

Defective Data

(Binomial)

Varying

Subgroup

Size

Constant

Subgroup

Size

Varying

Subgroup

Size

Constant

Subgroup

Size

u Chart c Chart p Chart np Chart

55

Improvement

• Control charts are one of many variation reduction

tools

• Controls charts detect change of the output variable

(y)

• The output changes because a critical input variable

(x) has changed

• Control charts provide clues that can help to identify

these critical inputs (x’s)

56

Clues to Discovering Critical x’s

• When did the change occur?

• What patterns are emerging?

� Shifts

• Gradual or Sudden?

� Trends

• Increasing or Decreasing?

� Unusual patterns or cycles?

57

Identification of Critical x’s

• To determine the critical x, i.e., the input causing

the shift, we need to consider:

� Delayed detection

� Multiple inputs causing shifts

� Lack of information on inputs

• We can also use screening experiments, scatter

diagrams, … to determine critical x’s

58

Transmission of Variation, y = f(x)

• Control charts can help to discover critical x’s that

are causing the process to shift

• Tighter control of these critical x’s will make the

process more stable

OUTPUT

INPUT

Relationship BetweenInput and Output

Variation of Input

TransmittedVariation

59

Transactional Improvement Process

� Select Project

� Define Project

Objective

� Form the Team

� Map the Process

� Identify Customer Requirements

� Identify Priorities

� Update Project File

� Select Project

� Define Project

Objective

� Form the Team

� Map the Process

� Identify Customer Requirements

� Identify Priorities

� Update Project File

� Control Critical x ’s

� Monitor y’s

� Validate Control Plan

� Identify further opportunities

� Close Project

15 20 25 30 35

LSL USL

Phase Review

1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

y

Phase Review

� Develop Detailed Process Maps

� Identify Critical Process Steps (x ’s) by looking for:

– Process Bottlenecks

– Rework / Repetition

– Non-value Added Steps

– Sources of Error / Mistake

� Map the Ideal Process

� Identify gaps between current and ideal

START

PROCESSSTEPS

DECISION

STOP

Phase Review

� Brainstorm Potential Improvement Strategies

� Select Improvement Strategy

� Plan and Implement Pilot

� Verify Improvement

� Implement Countermeasures

Criteria A B C D

Time + s - +

Cost + - + s

Service - + - +

Etc s s - +

15 20 25 30 35

LSL USL

Phase Review

Analyse Improve ControlMeasureDefine

Phase Review

Define

� Define Measures (y ’s)

� Check Data Integrity

� Determine Process Stability

� Determine Process Capability

� Set Targets for Measures

15 20 25 30 35

LSL USL

15 20 25 30 35

LSL USL

Phase Review

1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

y

Phase Review

� Develop Detailed Process Maps

� Identify Critical Process Steps (x ’s) by looking for:

– Process Bottlenecks

– Rework / Repetition

– Non-value Added Steps

– Sources of Error / Mistake

� Map the Ideal Process

� Identify gaps between current and ideal

START

PROCESSSTEPS

DECISION

STOP

Phase Review

� Brainstorm Potential Improvement Strategies

� Select Improvement Strategy

� Plan and Implement Pilot

� Verify Improvement

� Implement Countermeasures

Criteria A B C D

Time + s - +

Cost + - + s

Service - + - +

Etc s s - +

15 20 25 30 35

LSL USL

Phase Review

Analyse Improve ControlMeasureDefine

Phase Review

Define

� Define Measures (y ’s)

� Check Data Integrity

� Determine Process Stability

� Determine Process Capability

� Set Targets for Measures

60

Control Charts - Summary

• Charts can be constructed for variable or attribute data

• I-MR Charts should always be considered for attribute data

• Control Charts are used during capability studies to

determine process stability

• Real-Time control charts are used to detect shifts so that

causes of shifts can be identified and eliminated

• Should be used to control critical x’s (process input

variables) in order to reduce variability in process outputs

(y’s).

• Used to monitor y’s on an ongoing basis to ensure that the

process remains in control.