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Copyright 2012 BSI. All rights
reserved.
Analysis of Variance (ANOVA)
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Analysis of Variance - Learning Objectives
At the end of this session, delegates will be able to:
Understand the concept of additivity of variance
Interpret results from an Analysis of Variance Table Test results from Analysis of Variance for statistical significance
Copyright 2012BSI. All rightsreserved.
2
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Analysis of Variance - Agenda
1. Introduction
2. Calculation of Total Variance
3. One-Way Analysis of Variance
4. Components of Variation
5. Calculation of Sum of Squares
6. Degrees of Freedom
7. Calculation of the Mean Square
Copyright 2012BSI. All rightsreserved.
3
8. F-Ratio9. Testing for Significance
10. Estimating Components of Variation
11. Crossed v Nested Designs
12. Summary
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Define ImproveMeasure Control Control Critical xs
Monitor ys
1 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
Effect
C1 C2
C4
C3
C6C5
Select Project
Define ProjectObjective
Form the Team
Map the Process
Define Measures (ys)
Evaluate Measurement
System
DMAIC Improvement Process
Copyright 2012BSI. All rightsreserved.
4
Validate ControlPlan
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
2 1 1 1 2 2 2 2
31 2 2 1 1 2 2
4 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 CustomerRequirements
Identify Priorities
Update Project File
Phase Review
Stability Determine Process
Capability
Set Targets forMeasures
15 20 25 30 35
LSL USL
Phase Review
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Introduction to ANOVA
Analysis of Variance (ANOVA) is a technique that separates variation into individual components
These components of variation can then be interpreted to determine their importance
In Six Sigma, Analysis of Variance will be used to interpret:
Results from Gauge R&R Studies
Data from Designed Experiments
Results from Regression Analysis
Copyright 2012BSI. All rightsreserved.
5
Data from Response Surface Methodology
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Additivity of Variance
Although the standard deviation () is the most commonly used measure of variation, it is the
variance (2
) that is studied in Analysis of Variance
The reason for this is that variances are additive, which means that the total variation can be
divided into constituent parts
Copyright 2012BSI. All rightsreserved.
6
The following example demonstrates how this is done
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Data - Dimension
1 2 3 4 5
49.6 48.3 48.6 49.3 51.849.4 48.5 51.2 49.6 51.6
51.1 51.5 49.5 51.1 49.5
The data opposite represents the dimension of an
injection moulded component, with a nominal
Cavity Number
Copyright 2012BSI. All rightsreserved.
7
49.1 49.2 48.9 48.5 51.2
48.9 50.2 48.4 50.0 51.8
50.1 50.2 49.4 50.6 52.3
50.9 50.2 50.4 50.4 50.9
50.0 51.0 48.4 52.0 51.2
49.7 49.5 48.2 50.5 50.0
dimension of 50mm.
10 samples have been taken from each of 5
mould cavities.
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Data-Dimension
5251504948
A nderson-Darling Normality Test
V ariance 1.268
Skewness 0.05595
Kurtosis -1.00052
N 50
M inimum 48.200
A -Squared
1st Q uartile 49.275
M edian 50.150
3rd Q uartile 51.100
Maximum 52.300
95% C onfidence Interv al for Mean
0.41
P-V alue 0.323
Mean 50.110
StDev 1.126
Summary for data
Copyright 2012 BSI. All rights
reserved.8
Median
Mean
50.5050.2550.0049.7549.50
49.790 50.430
95% C onfidence Interva l for Median
49.567 50.500
95% C onfidence Interval for StDev
0.941 1.403
95% Confidence Intervals
We might want to investigate if there is a significant difference between cavities
We can use One-way ANOVA to investigate this
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The equation for the sample variance of the 50 observations can be calculated as follows:
Total Variance Calculation
)()(
222
2
====
====
ny
yyys
Copyright 2012 BSI. All rights
reserved.9
Where:
y = The sum of the individual observations
y2 = The sum of squares of the individual observations n = the number of observations
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Total Variance Calculation
1 2 3 4 5
49.6 48.3 48.6 49.3 51.8
49.4 48.5 51.2 49.6 51.6
51.1 51.5 49.5 51.1 49.5
51.6 51.8 50.5 48.7 50.249.1 49.2 48.9 48.5 51.2
48.9 50.2 48.4 50.0 51.8
50.1 50.2 49.4 50.6 52.3
Copyright 2012 BSI. All rights
reserved.10
5.25050.50.............1.514.496.49y=++++=
73.1256120.50.............1.514.496.49y 22222 =++++=
0.9 0. 0. 0. 0.9
50.0 51.0 48.4 52.0 51.2
49.7 49.5 48.2 50.5 50.0
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(((( ))))
1
2
2
22
1
======== n
n
yy
sn
(((( ))))5.2505 2
Total Variance Calculation
Copyright 2012BSI. All rightsreserved.
11
The total variance of our 50 observations is 1.268
The square root of the variance, the standard deviation, is 1.126
(These calculations can be performed simply in Minitab!)
268.149
50.221 ====
========
sn
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One-Way ANOVA
A one-way analysis of variance requires the following:
1. Identification of the Sources (Components) of Variation
2. Calculation of the Sum of Squares due to each Source of Variation
Copyright 2012BSI. All rightsreserved.
12
.
4. Calculation of the Mean Squares
5. Calculation of the F-Ratio
6. Test the statistical significance of the F-Ratio
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1. Components of Variation
Analysis of Variance (ANOVA) allows the decomposition of the total variance into its constituent
parts:
The components of variance in this study are:
2Between Cavity = Variation due to the different cavities
Copyright 2012 BSI. All rightsreserved.
13
(Between Cavities)
2Within Cavity = Variation due to the parts within each cavity
(Between Parts within Cavities)
2Total = 2Between Cavity +
2Within Cavity
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The total sum of squares is calculated as follows:
Strictl s eakin the sum of s uares is the sum of s uares around the mean known as the
(((( ))))(((( ))))
n
y
yyySSTotal
2
22 ========
2. Calculation of the Sum of Squares
Copyright 2012BSI. All rightsreserved.
14
corrected sum of squares. We always use the corrected sum of squares when estimating
variation.
(((( )))) (((( )))) 12.6250
5.250573.125612
22
2============
n
yySSTotal
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The sum of squares due to cavities is calculated as follows:
(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( ))))
(((( )))) (((( )))) (((( )))) (((( ))))5.2505)5.510()7.500(5.4934.5004.500 222222
22
5
2
4
2
3
2
2
2
1
++++++++++++++++
====
++++++++++++++++
==== p
ityBetweenCavn
yn
CCCCCSS
2. Calculation of the Sum of Squares
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C1, C2, C3, C4, C5 are the Sums for each Cavity (i.e the Sum of the 10 parts made on each cavity).
np is the number of parts made within each cavity
72.1461.12555033.125565
5010
========ityBetweenCav
ye ween av
SS
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The sum of squares due to within cavity variation can be calculated by subtraction since:
12.62====
++++====
Total
tiesWithinCaviitiesBetweenCavTotal
SS
SSSSSS
2. Calculation of the Sum of Squares
Copyright 2012BSI. All rightsreserved.
16
40.4772.1412.62
72.14
========
====
tiesWithinCavi
itiesBetweenCav
SS
SS
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Source of Variation
Between Cavities
Sum of Squares
14.72
2. Calculation of the Sum of Squares
Copyright 2012BSI. All rightsreserved.
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Within Cavities
Total
47.40
62.12
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3. Degrees of Freedom
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:
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Person 1 v Person 2
Tallest v Person 3
We would then know who the tallest person is.
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3. Rules for Degrees of Freedom
The following rules apply to Degrees of Freedom for One-Way ANOVA:
DF for Between Factor Variation = (Number of levels) 1
(Between Cavities)
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DF for Within Factor Variation = (Factor levels) x (Repeats 1)
(Within Cavities)
Total DF = (Number of Individual results 1)
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Source of Variation
Between Cavities
Sum of Squares
14.72
Degree of Freedom
4
3. Degrees of Freedom
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Total
.
62.12 49
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The Mean Square is calculated as follows:
Source of Variation Sum of Squares DF Mean Square
4. Calculation of the Mean Square
Mean Square = (Sum of Squares) / (Degrees of Freedom)
Copyright 2012BSI. All rightsreserved.
21
Between Cavities
Within Cavities
Total
14.72
47.40
62.12
4
45
49
3.680
1.053
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F-Ratio
3.49
Source of Variation
Between Cavities
Within Cavities
Total
Sum of Squares
14.72
47.40
62.12
DF
4
45
49
Mean Square
3.680
1.053
5. Calculation of the F-Ratio
Copyright 2012BSI. All rightsreserved.
22
The F-Ratio is used to test the significance of the between cavity variation. The higher the F-Ratio,
the more likely that the source of variation is significant.
F-ratio = (MSBetween Cavities) / (MSWithin Cavities) = 3.680/1.053 = 3.49
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We can test the significance of the between cavity variation by examining the F-Ratio, (named after
Sir Ronald Aylmer Fisher).
Examining the F tables for F0.05,4,45 gives a value of 2.61 (this is
6. Testing for Significance
49.3053.1680.3
SquareMeanCavityWithinSquareMeanCavityBetweenratio-F ============
Copyright 2012 BSI. All rights
reserved.23
nearest in the tables to F0.05,4,40)
Our value of 3.49 is greater than 2.61 so we can assume that the
between cavity variation is statistically significant.
This is equivalent to saying that there is a statistically significant
difference in dimension dependent on the cavity used.
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Open Worksheet: ANOVA-One-Way Unstacked
ANOVA - Minitab
Copyright 2012 BSI. All rights
reserved.24
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ANOVA - Minitab
Select: 1-5
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Analysis of Variance - Minitab
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26
The p-value gives the probability of no difference between the cavity averages.
As p = 0.014
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95% Confidence Intervals
Copyright 2012 BSI. All rights
reserved.27
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Crossed v Nested Designs
In the previous example we were investigating the effect of a single factor (cavity) on component
dimension
In many cases we might want to investigate two or more factors e.g. machine or operator
Depending upon how the levels of one factor appear with the levels of the other factor, we can say
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In Crossed Designs every level of one factor occurs with every level of another factor
In Nested Designs not all levels of one factor occur with every level of another factor
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Machine 1
Machine 2Factor A:- Machine
Crossed Design
Copyright 2012BSI. All rightsreserved. 29
Factors are CROSSED because each operator works on each machine
For example, when doing a Gauge R&R every component is measured by every operator so we
use a Crossed analysis
-
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Machine 1
Machine 2
Factor A:- Machine
Factor B:-
Nested Design
Copyright 2012BSI. All rightsreserved. 30
Factors are NESTED because different operators work on each machine
Operator
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Analysis of Variance - Summary
Analysis of Variance (ANOVA) is a technique that separates all variation into individual
components
These components of variation can then be interpreted to determine their importance
Analysis of Variance is a complicated subject and the advice of a statistician should be sought
Copyright 2012BSI. All rightsreserved.
31
before attempting any complex analyses
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Define ImproveMeasure Control Control Critical xs
Monitor ys
1 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
Effect
C1 C2
C4
C3
C6C5
Select Project
Define ProjectObjective
Form the Team
Map the Process
Define Measures (ys)
Evaluate MeasurementSystem
DMAIC Improvement Process
Copyright 2012BSI. All rightsreserved. 32
Validate ControlPlan
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
2 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
x
xx
x
xx
x
xx
x
Identify Customer
Requirements
Identify Priorities
Update Project File
Phase Review
Stability
Determine ProcessCapability
Set Targets forMeasures
15 20 25 30 35
LSL USL
Phase Review