5 Ssgb Amity Bsi Anova

7/29/2019 5 Ssgb Amity Bsi Anova

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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


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


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


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


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


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


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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


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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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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


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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



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


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


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



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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



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



17

Within Cavities

Total

47.40

62.12


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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



20

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)


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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


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


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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


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ANOVA - Minitab

Select: 1-5

Copyright2012BSI. All rightsreserved. 25


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Analysis of Variance - Minitab


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


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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

Copyright2012BSI. All rightsreserved.

28

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


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


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


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

5 Ssgb Amity Bsi Anova

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