Module-6
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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
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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?
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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
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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
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Measurement is accurate and precise
precise accurate
Measurement
Accuracy
Bias
Linearity
Measurement System Variation
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MeasurementSystemVariation
Reproducibility
Repeatability
Stability
Precision
Bias
Bias
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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
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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
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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
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Repeatability is the variation between successive measurements of the same part, samecharacteristic, by the same person using the same gauge.
Reproducibility
Reproducibility
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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
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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
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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
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Dimension
?
Variable Gauge R&R
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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)
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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
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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
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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
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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
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measurements Individual measurements
In Minitab the data is entered in single columns
Variable Gauge R&R - Minitab
Stat>Quality Tools>Gage Study>Gage R& R (Crossed)Enter Part, Operator, MeasurementCheck ANOVA Method
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Variable Gauge R&R - Minitab
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
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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
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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!
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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
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%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
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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
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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
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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
Two-Way ANOVA Table With Interaction
Variable Gauge R&R - Minitab
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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
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
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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
Components of Variation
120
100
% Contribution
% Study Var
% Process
% Tolerance
Gage name:
Date of study :
Reported by :
Tolerance:
Misc:
Components of Variation
Gage R&R (ANOVA) for Measurement
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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
Gage R&R (ANOVA) for Measurement
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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
Gage R&R (ANOVA) for Measurement
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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
Gage R&R (ANOVA) for Measurement
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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
Gage R&R (ANOVA) for Measurement
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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
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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
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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)
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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.
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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
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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
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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
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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.
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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.
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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:
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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
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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
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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
Attribute Gauge R&R - Results
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
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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
Attribute Gauge R&R - Results
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
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Appraiser
P
e
r
c
e
n
t
CBA
90
85
80
Appraiser
P
e
r
c
e
n
t
CBA
90
85
80
Attribute Gauge R&R - Results
The target effectiveness is always 100%
Possible Corrective Actions include:
Operator Training
Clarification of Standards
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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)
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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
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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
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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
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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
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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
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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
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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
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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
Calculation of the Sum of Squares
Copyright 2012 BSI. All rights reserved.
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
Calculation of the Sum of Squares
Copyright 2012 BSI. All rights reserved.
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
Calculation of the Sum of Squares
Copyright 2012 BSI. All rights reserved.
61
0387.01037.00480.00587.22491.2ityRepeatabil ==SS
Source of Variation
Between PartsBetween OperatorsOperator x Part
Sum of Squares
2.05870.04800.1037
Calculation of the Sum of Squares
Copyright 2012 BSI. All rights reserved.
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
Copyright 2012 BSI. All rights reserved.
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
Copyright 2012 BSI. All rights reserved.
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
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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
Copyright 2012 BSI. All rights reserved.
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
Copyright 2012 BSI. All rights reserved.
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
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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
Expected Mean Square
2
2ityRepeatabil
2PartOperator
2ityRepeatabil
2PartOperator
2Operator
2ityRepeatabil
2PartOperator
2Part
2
220
26
+
++
++
Source
Parts
Operators
Operator x Part
Repeatability
Estimating Components of Variation
Copyright 2012 BSI. All rights reserved.
69
0.0013 2 ityRepeatabilRepeatability
00225.0
0045.00013.00058.02
0058.02
0013.0
2PartOperator
2PartOperator
2ityRepeatabil
2PartOperator
2ityRepeatabil
=
==
=+
=
Estimating Components of Variation
Mean Square
0.2287
0.0240
0.0058
Expected Mean Square
2ityRepeatabil
2PartOperator
2ityRepeatabil
2PartOperator
2Operator
2ityRepeatabil
2PartOperator
2Part
2
220
26
+
++
++
Source
Parts
Operators
Operator x Part
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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
Expected Mean Square
2ityRepeatabil
2PartOperator
2ityRepeatabil
2PartOperator
2Operator
2ityRepeatabil
2PartOperator
2Part
2
220
26
+
++
++
Source
Parts
Operators
Operator x Part
Estimating Components of Variation
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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
=
=
=
=
Estimating Components of Variation
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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
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73
MeasurementSystemVariation0.00446
Repeatability0.00130
Operatorby part
Interaction0.00225
Variances are additive!