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Module-6

Copyright 2012 BSI. All rights reserved.

Module-6Measurements Systems Analysis

Measurements Systems Analysis - Agenda

1. Is our data accurate?

Repeatability & Reproducibility

Accuracy & Precision

DMAIC

Measurements System Variation

Bias, Linearity, Stability, Repeatability, Reproducibility, Calibration, Gauge R&R

2. Variable Gauge R&R


2

Parts, Operators, Variation

Is the gauge good?

Workshop

3. Attribute Gauge R&R

Workshop

4. Appendix

Analysis of Variance (ANOVA)

Gauge R & R is a means of assessing the repeatability and reproducibility of our measurement systems.

Gauge R & R studies are carried out in order to discover how much of the process variation is due to the measurement device and measurement methods.

Is our Data Accurate?


3

Dimension

?

Define ImproveMeasure Control Control Critical xs

Monitor ys1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

Analyse Characterise xs

Optimise xs

y=f(x1,x2,..)

y

x

. . .

. . .

. .

. . .

. . .

Identify Potential xs

Analyse xs

Run 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1

Effect

C1 C2

C4

C3

C6C5

Select Project Define Project

Objective Form the Team

Map the Process

Define Measures (ys)

Evaluate Measurement System

Determine Process

DMAIC Improvement Process


4

Validate Control Plan

Close Project

y

Phase Review

Set Tolerances for xs Verify Improvement

15 20 25 30 35

LSL USL

Phase Review

Select Critical xs

Phase Review

1 1 1 1 1 1 1 12 1 1 1 2 2 2 23 1 2 2 1 1 2 24 1 2 2 2 2 1 15 2 1 2 1 2 1 26 2 1 2 2 1 2 17 2 2 1 1 2 2 18 2 2 1 2 1 1 2

x

xx

xx

xx

xx

x

x

Identify Customer Requirements

Identify Priorities Update Project File

Phase Review

Determine Process Stability

Determine Process Capability

Set Targets for Measures

15 20 25 30 35

LSL USL

Phase Review

Measurement is accurate but not precise

Measurement is precise but not accurate

Measurement Accuracy & Precision


5

Measurement is accurate and precise

precise accurate

Measurement

Accuracy

Bias

Linearity

Measurement System Variation


6

MeasurementSystemVariation

Reproducibility

Repeatability

Stability

Precision

Bias

Bias


7

ObservedAverage

TrueValue

Bias is the difference between the observed average of the measurements and the true value.

Me

a

s

u

r

e

d

V

a

l

u

e Non-Linearity

Gauge is measuring lower than true value at high end

Linearity


8

Linearity is the difference in bias values over the expected operating range of the measurement gauge.

Reference Value

M

e

a

s

u

r

e

d

V

a

l

u

e

Stability

Stability


9

Time2

Time1

Stability is the variation (differences) in the average over extended periods of time usingthe same gauge and appraiser to repeatedly measure the same part

Repeatability

Repeatability


10

Repeatability is the variation between successive measurements of the same part, samecharacteristic, by the same person using the same gauge.

Reproducibility

Reproducibility


11

Operator2

Operator1

Reproducibility is the difference in the average of the measurements made by differentpeople using the same instrument when measuring the identical characteristic on thesame pieces.

Measurement System Variation

Accuracy

Stability

Bias

Linearity Calibration


12

Reproducibility

Repeatability

Stability

Precision Gauge R&R

Calibration

The Bias of a gauge can be assessed by repeat measurements of a known reference unit

This can be extended across the operating range of the gauge in a Gauge Linearity Study

The Stability of the gauge can be assessed by control charting a reference unit

Should not routinely recalibrate, instead if reference unit tests outside the control limits, then


13

Should not routinely recalibrate, instead if reference unit tests outside the control limits, then

re-calibrate

If measurement device requires frequent recalibration, attempt to improve stability

Gauge R & R is a means of assessing the repeatability and reproducibility of our measurement systems.

Gauge R & R studies are carried out in order to discover how much of the process variation is due to the measurement device and measurement methods.

Gauge R & R


14

Dimension

?

Variable Gauge R&R


Variable Gauge R&R

Variable Gauge R&R

Requirements:

A minimum of two operators (recommend 3 or 4)

At least 10 parts which should be chosen to represent the full range of manufacturing variation

(it may be acceptable to use fewer parts in some special cases)


16

Part 1

Part 4

Part 2

Part 3

Part 10

Part 5

Each part should be measured two or three times in a random order

Operators should not be aware of the previous result when measuring the same part

There are two methods available:

1. Analysis of Variance (ANOVA)

2. X-Bar and R

Variable Gauge R&R


17

The ANOVA method is:

the recommended approach

takes into account any interactive effect between operator and part

Reproducibility

Part-to-PartVariation

Operator

Operator

Variable Gauge R&R

OverallVariation


18

MeasurementSystemVariation

Repeatability

Operatorby part

Interaction

We want the Part-to-Part component to be large!

Part Operator 1 Operator 2 Operator 31 0.65 0.55 0.501 0.60 0.55 0.552 1.00 1.05 1.052 1.00 0.95 1.003 0.85 0.80 0.803 0.80 0.75 0.804 0.85 0.80 0.804 0.95 0.75 0.805 0.55 0.40 0.45

Variable Gauge R&R - Example


19

5 0.55 0.40 0.455 0.45 0.40 0.506 1.00 1.00 1.006 1.00 1.05 1.057 0.95 0.95 0.957 0.95 0.90 0.958 0.85 0.75 0.808 0.80 0.70 0.809 1.00 1.00 1.059 1.00 0.95 1.0510 0.60 0.55 0.8510 0.70 0.50 0.80

Open Worksheet: Gauge R&R

The Part numbers being measured

Operators performing measurements

Each operator measures each part twice

Individual measurements

Variable Gauge R&R - Minitab


20

measurements Individual measurements

In Minitab the data is entered in single columns


Stat>Quality Tools>Gage Study>Gage R& R (Crossed)Enter Part, Operator, MeasurementCheck ANOVA Method


21


Two-Way ANOVA Table With Interaction

Source DF SS MS F PPart 9 2.05871 0.228745 39.7178 0.000Operator 2 0.04800 0.024000 4.1672 0.033Part * Operator 18 0.10367 0.005759 4.4588 0.000Repeatability 30 0.03875 0.001292Total 59 2.24913

Gage R&R


22

Gage R&R

%ContributionSource VarComp (of VarComp)Total Gage R&R 0.0044375 10.67Repeatability 0.0012917 3.10Reproducibility 0.0031458 7.56

Operator 0.0009120 2.19Operator*Part 0.0022338 5.37

Part-To-Part 0.0371644 89.33Total Variation 0.0416019 100.00

p

Variance Component Estimates

OverallVariation0.0416019 Reproducibility

0.0031458

Part-to-PartVariation0.0371644 Operator

0.0009120

Operator


23

Repeatability0.0012917

MeasurementSystemVariation0.0044375

Operatorby part

Interaction0.0022338

Variances are additive!

Variable Gage R&R Standard Deviations

Study Var %Study Var

Source StdDev (SD) (6 * SD) (%SV)Total Gage R&R 0.066615 0.39969 32.66

Repeatability 0.035940 0.21564 17.62

This is the gauge standard deviation, R&R = 0.066615Remember that standard deviations are not additive!


24

Repeatability 0.035940 0.21564 17.62

Reproducibility 0.056088 0.33653 27.50

Operator 0.030200 0.18120 14.81

Operator*Part 0.047263 0.28358 23.17

Part-To-Part 0.192781 1.15668 94.52

Total Variation 0.203965 1.22379 100.00

We would like the total measurement system variation (Gauge R&R) to be as small as possible. Calculate the percentage of the process tolerance taken up by the measurement system variation, represented by 6 x the gauge standard deviation. This is known as %Precision/Tolerance or %P/T.

Interpreting the Results


25

%P/T.

The Process Tolerance is equivalent to the difference between the upper and lower specification limits (USL LSL).

Tolerance Process6%100%P/T R&R=

Is the Gauge Good?

% P/T(6R&R/Process Tolerance)

Acceptability

0 - 10% Very Good (Six Sigma Gauge)

10 - 30% May be Acceptable


26

The interpretation will also depend on the current level of process variation

>30% Probably Not Acceptable

Note that these guidelines are as recommended in Measurement Systems Analysis Third Edition published in March 2002 as part ofQS-9000 and developed in conjunction with AIAG.

Is the Gauge Good?

% R&R

If the %P/T is greater than 10%, then a secondary calculation can be used to decide whether the gauge can be used during the DMAIC activity.

Comparing R&R to the current process variation indicates whether the measurement device is currently causing a problem. This is known as %R&R.

We need an independent estimate of the process (total) variation (the value from the Gauge


27

We need an independent estimate of the process (total) variation (the value from the Gauge R&R is based on only a few samples)

We would like the measurement standard deviation to be less than the total standard deviation

50% 100%R&%R

tal)Process(to

R&R

1. Comparing the gauge variation to the process tolerance:

This is greater than 10% so the gauge will not be good enough for six sigma. As the process improves the gauge will become a problem. To improve this gauge we should start by addressing the reproducibility.

28.5%100%1.40.06666

100%Tolerance

%P/T 6 R&R ===

Interpreting the Results


28

addressing the reproducibility.

2. Comparing the gauge variation to the process variation:

This is less than 50% so the gauge is not the limiting factor at the moment. We can use this gauge for process improvement.

37%100%0.18

0.0666 100%R&%R

tal)Process(to

R&R===

Open Worksheet: GaugeR&RStat>Quality Tools>Gage Study>Gage R& R (Crossed)Enter Part, Operators, MeasurementCheck ANOVA MethodSelect Options: Study Variation: 6

Process Tolerance: 1.4Historical standard deviation: 0.18




29


Source DF SS MS F PPart 9 2.05871 0.228745 39.7178 0.000Operator 2 0.04800 0.024000 4.1672 0.033Part * Operator 18 0.10367 0.005759 4.4588 0.000Repeatability 30 0.03875 0.001292Total 59 2.24913

Components of Variation

Components of VariationGage R&R

%ContributionSource VarComp (of VarComp)Total Gage R&R 0.0044375 10.67

Repeatability 0.0012917 3.10Reproducibility 0.0031458 7.56

Operator 0.0009120 2.19Operator*Part 0.0022338 5.37

Part-To-Part 0.0371644 89.33Total Variation 0.0416019 100.00


30

Study Var %Study Var %Tolerance %ProcessSource StdDev (SD) (6 * SD) (%SV) (SV/Toler) (SV/Proc)Total Gage R&R 0.066615 0.39969 32.66 28.55 37.01

Repeatability 0.035940 0.21564 17.62 15.40 19.97Reproducibility 0.056088 0.33653 27.50 24.04 31.16

Operator 0.030200 0.18120 14.81 12.94 16.78Operator*Part 0.047263 0.28358 23.17 20.26 26.26

Part-To-Part 0.192781 1.15668 94.52 82.62 107.10Total Variation 0.203965 1.22379 100.00 87.41 113.31

Number of Distinct Categories = 4


120

100

% Contribution

% Study Var

% Process

% Tolerance

Gage name:

Date of study :

Reported by :

Tolerance:

Misc:


Gage R&R (ANOVA) for Measurement


31

P

e

r

c

e

n

t

Part-to-PartReprodRepeatGage R&R

100

80

60

40

20

0

% Tolerance

Part to Part Measurements

1.1

1.0

Gage name:

Date of study :

Reported by :

Tolerance:

Misc:

Measurement by Part



32

Part

10987654321

0.9

0.8

0.7

0.6

0.5

0.4

Operator by Part Interaction

1.1

1.0

Operator

1

2

3

Gage name:

Date of study :

Reported by :

Tolerance:

Misc:

Operator * Part Interaction



33

Part

A

v

e

r

a

g

e

10 9 8 7 6 5 4 3 2 1

0.9

0.8

0.7

0.6

0.5

0.4

3

Measurements by Operator

1.1

1.0

Gage name:

Date of study :

Reported by :

Tolerance:

Misc:

Measurement by Operator



34

Operator

321

0.9

0.8

0.7

0.6

0.5

0.4

Xbar and R Chart by Operator

M

e

a

n

1.0

0.8

__X=0.8075UCL=0.8796

LCL=0.7354

1 2 3

Gage name:

Date of study :

Reported by :

Tolerance:

Misc:

Xbar Chart by Operator



35

S

a

m

p

l

e

R

a

n

g

e

0.12

0.08

0.04

0.00

_R=0.0383

UCL=0.1252

LCL=0

1 2 3

S

a

m

p

l

e

M

0.8

0.6

0.4

X=0.8075LCL=0.7354

R Chart by Operator

Rounding Errors

Rounding is another component of measurement variation which needs to be minimised

It can be shown that to avoid rounding error getting in the way of achieving six sigma quality, it

is necessary to have a minimum of 14 discrete values between the upper and lower specification

For one-side specifications, there need to be at least 7 discrete values between the process


36

average and the specification limit

UG 37

Rounding Errors - Interpolating

If possible interpolate between graduation marks

For example, thermometers are frequently marked to the nearest degree but can be read to the nearest 0.2 degrees, even if the last digit is not entirely accurate

Interpolating frequently reduces and never increases the measurement variation


37

Improving the Measurement System

Gauge incapable:

Repeatability (Gauge)

Take multiple measurements and use average (short term fix)

Mistake proofing (e.g. provision of tooling to hold part during measurement)

May need maintenance

Reproducibility (Operators)


38

Use 1 operator (short term fix during improvement only)

Have several operators measure the part and take the average (short term fix)

Ensure consistency (training, SOPs, WIS, )

Mistake proofing (e.g. provision of tooling to hold part during measurement)

Calibrations on the gauge dial may not be clear

Reproducibility Operator x Part Interaction

Identify cause of interaction and then as Operator

Destructive Gauge R&R

Destructive gauge testing means that it is impossible to carry out repeat tests!

To complete an assessment of a destructive gauge it is therefore necessary to assume homogeneity within batches.

If there is much more difference in parts between batches than within batches, then a standard variable Gauge R & R may be sufficient.


39

Workshop Variable Gauge R&R

Using the provided measuring device and products carry out a Gauge R&R

Use three operators and measure each part twice

Ensure that the order of measuring is randomised

Analyse the data using Minitab


40

Analyse the data using Minitab

What could you do, if anything to improve the Measurement System?

Prepare a short report detailing your findings

Attribute Gauge R&R


Attribute Gauge R&R

Attribute Gauge R&R

A Gauge R&R study can also be carried out on attribute data

Using attribute data, we would have a problem with the measurement system if:

Operators disagree with each others evaluation of a piece

The same operator gains different results from a repeat evaluation of the same piece


42

Attribute Measurement System

An attribute measurement system compares each part to a standard and either accepts or rejects

the part.

The screen effectiveness is the ability of the attribute measurement system to properly

discriminate good from bad.


43

Screen effectiveness of 100% is desirable.

Conducting Attribute Gauge R&R

1. Select a minimum of 30 parts from the process. These parts should represent the full

spectrum of process variation (good parts, defective parts, borderline parts).

2. An expert inspector performs an evaluation of each part, classifying it as Good or Not

Good.


44

3. Independently and in a random order, each of 2 or 3 operators should assess the parts as

Good or Not Good.

4. Calculate effectiveness scores.

Attribute Gauge R&R

Column containing parts being assessed

Text column containing expert

assessment (can use words or numbers but

must be consistent)

Column containing parts being assessed

Text column containing expert

assessment (can use words or numbers but

must be consistent)

Open Worksheet: Attribute Gage R&RMinitab Data Layout:


45

Text column containing operator

performing measurements

Text column containing results of measurements (can

use words or numbers but must be consistent)

Text column containing operator

performing measurements

Text column containing results of measurements (can

use words or numbers but must be consistent)

Attribute Gauge R&R

Stat>Quality Tools>Attribute Agreement AnalysisEnter Results in Attribute Column, Part in Samples, Appraiser in Appraisers and Expert in Known standard/attributeClick on Results button and select Percentages


46

Attribute Gauge R&R - Results

Attribute Agreement AnalysisWithin AppraiserAssessment AgreementAppraiser # Inspected # Matched Percent (%) 95.0% CI A 30 28 93.3 ( 77.9, 99.2)B 30 30 100.0 ( 90.5, 100.0)C 30 30 100.0 ( 90.5, 100.0)# Matched: Appraiser agrees with him/herself across trials.

Appraiser A was not consistent on two out of

thirty parts inspected


47

Each Appraiser vs StandardAssessment Agreement

Appraiser # Inspected # Matched Percent (%) 95.0% CI A 30 28 93.3 ( 77.9, 99.2)B 30 29 96.7 ( 82.8, 99.9)C 30 29 96.7 ( 82.8, 99.9)# Matched: Appraiser's assessment across trials agrees with standard.

Appraiser A disagreed with expert on two parts,

Appraiser B and C disagreed with expert on

one part


Assessment Disagreement # Not Good/ # Good/

Appraiser Good Percent (%) Not Good Percent (%) # Mixed Percent (%) A 0 0.0 0 0.0 2 6.7 B 1 6.7 0 0.0 0 0.0 C 1 6.7 0 0.0 0 0.0 # Not Good/Good: Assessments across trials = Not Good / standard = Good.# Good/Not Good: Assessments across trials = Good / standard = Not Good.# Mixed: Assessments across trials are not identical.

Between AppraisersAssessment Agreement Appraiser A,B and C agreed

Appraiser B assessed one part as Not Good when the

standard (expert) assessed it as Good

Assessment Disagreement # Not Good/ # Good/

Appraiser Good Percent (%) Not Good Percent (%) # Mixed Percent (%) A 0 0.0 0 0.0 2 6.7 B 1 6.7 0 0.0 0 0.0 C 1 6.7 0 0.0 0 0.0 # Not Good/Good: Assessments across trials = Not Good / standard = Good.# Good/Not Good: Assessments across trials = Good / standard = Not Good.# Mixed: Assessments across trials are not identical.

Between AppraisersAssessment Agreement Appraiser A,B and C agreed

Appraiser B assessed one part as Not Good when the

standard (expert) assessed it as Good


48

Assessment Agreement# Inspected # Matched Percent (%) 95.0% CI

30 26 86.7 ( 69.3, 96.2)# Matched: All appraisers' assessments agree with each other.

All Appraisers vs StandardAssessment Agreement# Inspected # Matched Percent (%) 95.0% CI

30 26 86.7 ( 69.3, 96.2)# Matched: All appraisers' assessments agree with standard.

Appraiser A,B and C agreed on 26 out of 30 parts

inspected

Appraiser A,B and C all agreed with the standard on 26 out of 30 parts inspected

Assessment Agreement# Inspected # Matched Percent (%) 95.0% CI

30 26 86.7 ( 69.3, 96.2)# Matched: All appraisers' assessments agree with each other.

All Appraisers vs StandardAssessment Agreement# Inspected # Matched Percent (%) 95.0% CI

30 26 86.7 ( 69.3, 96.2)# Matched: All appraisers' assessments agree with standard.

Appraiser A,B and C agreed on 26 out of 30 parts

inspected

Appraiser A,B and C all agreed with the standard on 26 out of 30 parts inspected


100

95

95.0% C I

Percent100

95

95.0% C I

Percent

Date of study:

Reported by:

Name of product:

Misc:

Assessment Agreement

Within Appraisers Appraiser vs Standard


49

Appraiser

P

e

r

c

e

n

t

CBA

90

85

80

Appraiser

P

e

r

c

e

n

t

CBA

90

85

80


The target effectiveness is always 100%

Possible Corrective Actions include:

Operator Training

Clarification of Standards


50

Simplification of Standards

Conversion to Variable Data

Workshop Attribute Gauge R&R

From your team select two expert inspectors.

The experts should select 20 sweets, roughly half good (pass) and half bad (fail).

Some sweets should be borderline.

Carry out a Gauge R&R

Use two operators and measure each part twice (if more time available use three operators)


51

Use two operators and measure each part twice (if more time available use three operators)

Ensure that the order of measuring is randomised

Analyse the data

What could you do, if anything, to improve the Measurement System?

Prepare a short report detailing your findings.

Measurement Systems Analysis - Summary

Measurement errors can account for a large proportion of the variation in our measures (ys)

We must evaluate our measurement systems before assessing process stability or process

capability

Errors in measurement systems can come from a variety of sources


52

Errors in measurement systems can come from a variety of sources

Action should be taken to improve the capability of our measurement systems if they are found

to be inadequate

Define ImproveMeasure Control Control Critical xs

Monitor ys1 5 10 15 20

10.2

10.0

9.8

9.6

Upper Control Limit

Lower Control Limit

Analyse Characterise xs

Optimise xs

y=f(x1,x2,..)

y

x

. . .

. . .

. .

. . .

. . .

Identify Potential xs

Analyse xs

Run 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1

Effect

C1 C2

C4

C3

C6C5

Select Project Define Project

Objective Form the Team

Map the Process

Define Measures (ys)

Evaluate Measurement System

Determine Process

DMAIC Improvement Process


53

Validate Control Plan

Close Project

y

Phase Review

Set Tolerances for xs Verify Improvement

15 20 25 30 35

LSL USL

Phase Review

Select Critical xs

Phase Review

1 1 1 1 1 1 1 12 1 1 1 2 2 2 23 1 2 2 1 1 2 24 1 2 2 2 2 1 15 2 1 2 1 2 1 26 2 1 2 2 1 2 17 2 2 1 1 2 2 18 2 2 1 2 1 1 2

x

xx

xx

xx

xx

x

x

Identify Customer Requirements

Identify Priorities Update Project File

Phase Review

Determine Process Stability

Determine Process Capability

Set Targets for Measures

15 20 25 30 35

LSL USL

Phase Review

Appendix - ANOVA


Appendix - ANOVA

ANOVA Table - Construction

Construction of an Analysis of Variance (ANOVA) table requires the following:

1. Identification of the Sources (Components) of Variation

2. Calculation of the Sum of Squares due to each Source of Variation

3. Assignment of the appropriate Degrees of Freedom


55

3. Assignment of the appropriate Degrees of Freedom

4. Calculation of the Mean Squares

5. Calculation of the F-Ratio

Analysis of Variance (ANOVA) allows the decomposition of the variability in the Gauge R&R study.

The components of variation in the Gauge R&R study are:

2Part = Variation due to the different parts

1. Components of Variation


56

Part = Variation due to the different parts2Operator = Variation due to different operators2Operator x Part = Variation due to the interaction between operator

and part2Repeatability = Variation due to gauge repeatability2Total =

2Part +

2Operator +

2Operator x Part +

2Repeatability

The total sum of squares is calculated as follows:

Strictly speaking the sum of squares column is the sum of squares around the mean, known as the corrected sum of squares. We always use the corrected sum of squares when estimating variation.

( ) ( )n

yyyySSTotal

222

==

2. Calculation of the Sum of Squares


57

use the corrected sum of squares when estimating variation.

( ) ( ) 2491.21234.393725.416045.483725.41

3725.4180.0...........00.100.160.065.0

45.4880.0.............00.100.160.065.0

222

222222

====

=+++++=

=+++++=

n

yySS

y

y

Total

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )6045.48

600.4.........80.405.640.3

.........

22222

2210

23

22

21

++++=

++++=

Part

pPart

SS

n

yn

PPPPSS

The sum of squares due to parts is calculated as follows:

Calculation of the Sum of Squares


58

0587.21234.391821.41 ==PartSS

Where:P1, P2, P3..P10 are the Sums for each Parti.e the Sum of the 6 measurements made on each part.np is the number of individual measurements of each part.

The sum of squares due to operators is calculated as follows:

(((( )))) (((( )))) (((( )))) (((( ))))(((( )))) (((( )))) (((( )))) (((( ))))

6045.48

2055.1635.1555.16 2222

223

22

21

++++++++====

++++++++====

Operator

o

Operator

SS

n

yn

OOOSS



59

Where:O1, O2, O3 are the Sums for each Operatori.e the sum of the 20 measurements made by each operator.no is the number of measurements made by each operator.

0480.01234.391714.39 ========OperatorSS

The sum of squares due to the interaction between operators and parts is calculated as follows:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0587.20480.0

6045.48

265.1........00.225.1

..........

2222

22103

221

211

++=

++=

PartOperator

PartOperatorPO

PartOperator

SS

SSSSn

yn

POPOPOSS



60

Where:O1P1, O1P2,.O3P10 are the Sums for each Operator & Part Combinationi.e the sum of the 2 measurements made by each operator on each part.nOxP is the number of measurements made by each operator on each part.

1037.00587.20480.01234.393338.41

0587.20480.0602

==

PartOperator

PartOperator

SS

SS

The sum of squares due to repeatability is obtained by subtraction:

ityRepeatabil = SSSSSSSSSS PartOperatorOperatorPartTotal



61

0387.01037.00480.00587.22491.2ityRepeatabil ==SS

Source of Variation

Between PartsBetween OperatorsOperator x Part

Sum of Squares

2.05870.04800.1037



62

Operator x PartRepeatability

Total

0.10370.0387

2.2491

Degrees of Freedom is a statistical concept relating to the number of paired comparisons required to distinguish between items.

For example, we need to find the tallest person out of 3 people. 2 comparisons would be required:

3. Degrees of Freedom


63

people. 2 comparisons would be required:

Person 1 v Person 2Tallest v Person 3

We would then know who the tallest person is.

The following rules apply to Degrees of Freedom:

DF for a Factor (Main Effect) = (Number of Levels) 1

DF for interactions = Product of the DF of the Factors involved

Rules for Degrees of Freedom


64

DF for interactions = Product of the DF of the Factors involved

DF for Repeatability = (Product of Factor Levels) x (Repeats 1)

Total DF = (Number of Individual Results) - 1

Source of Variation

Between PartsBetween Operators

Sum of Squares

2.05870.0480

Degreesof

Freedom

92

Degrees of Freedom


65

Between OperatorsOperator x PartRepeatability

Total

0.04800.10370.0387

2.2491

21830

59

The Mean Square is calculated as follows:

Mean Square = (Sum of Squares) / (Degrees of Freedom)

Source of Variation Sum of Squares DF Mean Square

4. Calculation of the Mean Squares


66

Between PartsBetween OperatorsOperator x PartRepeatability

Total

2.05870.04800.10370.0387

2.2491

921830

59

0.22870.02400.00580.0013

Source of Variation

Between PartsBetween OperatorsOperator x PartRepeatability

Total

Sum of Squares

2.05870.04800.10370.0387

2.2491

DF

92

1830

59

Mean Square

0.22870.02400.00580.0013

F-Ratio

39.434.144.46

5. Calculation of the F-Ratio


67

Total 2.2491 59

The F-Ratio is used to test the significance of each source of variation. F-Ratio for Parts = (MSParts) / (MSOperator x Part)F-Ratio for Operators = (MSOperators) / (MSOperator x Part)F-Ratio for Operator x Part = (MSOperators x Parts) / (MSRepeatability)

Mean Square

The Mean Square column is expected to contain the following components of variation. This expected mean square is only applicable to this current study, where we have 3 operators, 10 parts and 2 repeat measurements. For other studies, the number of the components will change. (Fortunately, Minitab can do this for us!)

Expected Mean SquareSource

Estimating Components of Variation


68

Mean Square

0.2287

0.0240

0.0058

0.0013

Expected Mean Square

2ityRepeatabil

2ityRepeatabil

2PartOperator

2ityRepeatabil

2PartOperator

2Operator

2Repeatability

2PartOperator

2Part

2

220

26

+

++

++

Source

Parts

Operators

Operator x Part

Repeatability

Mean Square

0.2287

0.0240

0.0058

0.0013


2

2ityRepeatabil

2PartOperator

2ityRepeatabil

2PartOperator

2Operator

2ityRepeatabil

2PartOperator

2Part

2

220

26

+

++

++

Source

Parts

Operators

Operator x Part

Repeatability



69

0.0013 2 ityRepeatabilRepeatability

00225.0

0045.00013.00058.02

0058.02

0013.0

2PartOperator

2PartOperator

2ityRepeatabil

2PartOperator

2ityRepeatabil

=

==

=+

=


Mean Square

0.2287

0.0240

0.0058


2ityRepeatabil

2PartOperator

2ityRepeatabil

2PartOperator

2Operator

2ityRepeatabil

2PartOperator

2Part

2

220

26

+

++

++

Source

Parts

Operators

Operator x Part


70

00091.020

0013.0)00225.0(2)0240.0(

20240.020

0240.0220

2Operator

2ityRepeatabil

2PartOperator

2Operator

2ityRepeatabil

2PartOperator

2Operator

=

=

=

=++

0.00132

ityRepeatabilRepeatability

Mean Square

0.2287

0.0240

0.0058


2ityRepeatabil

2PartOperator

2ityRepeatabil

2PartOperator

2Operator

2ityRepeatabil

2PartOperator

2Part

2

220

26

+

++

++

Source

Parts

Operators

Operator x Part



71

0.00132

ityRepeatabilRepeatability

03715.06

0013.0)00225.0(22287.0

22287.06

2287.026

2Part

2ityRepeatabil

2PartOperator

2Part

2ityRepeatabil

2PartOperator

2Part

=

=

=

=++

00130.0

00225.0

00091.003715.0

2ityRepeatabil

2PartOperator

2Operator

2Part

=

=

=

=



72

04161.000130.000225.000091.003715.0

00130.0

2Total

2ityRepeatabil

2PartOperator

2Operator

2Part

2Total

ityRepeatabil

=+++=

+++=

=

We have established estimates of each of the components of variation!

OverallVariation0.04161 Reproducibility

0.00316

Part-to-PartVariation0.03715 Operator

0.00091

Operator

Variance Component Estimates


73

MeasurementSystemVariation0.00446

Repeatability0.00130

Operatorby part

Interaction0.00225

Variances are additive!

6 ssgb amity bsi msa p

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