Statistics and ANOVA
ME 470
Fall 2013
Here are some interesting on-the-spot designs
from the past and this class.
Fall 2009, 0.27 Cost/Height
Winner, Spring 2010
15” Tall
8$ Cost
0.533 Cost/Height
Fall 2011
Height = 12
Cost = 6
Cost/Height = 0.5
Fall 2011
Height = 24
Cost = 12
Cost/height = 0.5
I really enjoy the on-the-spot design.
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What did you learn about the design process?
There are many challenges in product development
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Trade-offs
Dynamics
Details
Time pressure
Economics
Design is a process
that requires
 Why do I love product development?making decisions.
 Getting something to work
 Satisfying societal needs
 Team diversity
 Team spirit
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Product Development Phases
Concept
System-Level
Development Design
Planning
Detail
Design
Testing and
Refinement
Production
Ramp-Up
Concept Development Process
Mission
Statement
Identify
Customer
Needs
Establish
Target
Specifications
Generate
Product
Concepts
Select
Product
Concept(s)
Test
Product
Concept(s)
You will practice the
entire concept
development
Perform Economic Analysis
process with
Benchmark Competitive Products
your group project
Build and Test Models and Prototypes
Set
Final
Specifications
Plan
Downstream
Development
Development
Plan
We will use statistics to
make good design decisions!
We will categorize populations by the mean,
standard deviation, and use control charts to
determine if a process is in control.
We may be forced to run experiments to
characterize our system. We will use valid
statistical tools such as Linear Regression,
DOE, and Robust Design methods to help us
make those characterizations.
Cummins asked a capstone group to investigate
improvements for turbo charger lubrication sealing.
5.9L High Output Cummins Engine
Cummins Inc. was dissatisfied with the integrity of
their turbocharger oil sealing capabilities.
Here are pictures of oil leakage.
Oil Leakage into Compressor Housing
Oil Leakage on Impellor Plate
The students developed four prototypes for testing. After testing, they wanted to know
which solution to present to Cummins. You will analyze their data to make a suggestion.
How can we use statistics to make sense
of data that we are getting?
Quiz for the day
 What can we say about our M&Ms?
 We will look at the results first and then you
can do the analysis on your own.
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Statistics can help us examine the data and draw
justified conclusions.
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What does the data look like?
What is the mean, the standard deviation?
What are the extreme points?
Is the data normal?
Is there a difference between years? Did one class get more
M&Ms than another?
If you were packaging the M&Ms, are you doing a good
job?
If you are the designer, what factors might cause the
variation?
Why would we care about this data in design?
If I am a plant manager, do I like one distribution better
than another?
How do we interpret the boxplot?
largest value excluding outliers
Boxplot of BSNOx
2.45
Q3
2.40
(Q2), median
BSNOx
2.35
2.30
Q1
2.25
2.20
outliers are marked as ‘*’
smallest value excluding outliers
Values between 1.5 and 3 times away from the middle 50% of the data are outliers.
This is a density description of the data.
The Anderson-Darling normality test is used to
determine if data follow a normal distribution.
If the p-value is lower than the predetermined level of significance, the data
do not follow a normal distribution.
Anderson-Darling Normality Test
Measures the area between the fitted line (based on chosen distribution) and the
nonparametric step function (based on the plot points). The statistic is a squared
distance that is weighted more heavily in the tails of the distribution. AndersonSmaller Anderson-Darling values indicates that the distribution fits the data better.
The Anderson-Darling Normality test is defined as:
H0: The data follow a normal distribution.
Ha: The data do not follow a normal distribution.
Another quantitative measure for reporting the result of the normality test is the p-value. A
small p-value is an indication that the null hypothesis is false. (Remember: If p is low, H0
must go.)
P-values are often used in hypothesis tests, where you either reject or fail to reject a null
hypothesis. The p-value represents the probability of making a Type I error, which is
rejecting the null hypothesis when it is true. The smaller the p-value, the smaller is the
probability that you would be making a mistake by rejecting the null hypothesis.
It is customary to call the test statistic (and the data) significant when the null hypothesis H0
is rejected, so we may think of the p-value as the smallest level α at which the data are
significant.
You can use the “fat pencil” test in addition to
the p-value.
Note that our p value is quite low, which makes us consider rejecting
the fact that the data are normal. However, in assessing the closeness
of the points to the straight line, “imagine a fat pencil lying along the
line. If all the points are covered by this imaginary pencil, a normal
distribution adequately describes the data.” Montgomery, Design
and Analysis of Experiments, 6th Edition, p. 39
If you are confused about whether or not to consider the data normal,
it is always best if you can consult a statistician. The author has
observed statisticians feeling quite happy with assuming very fat
lines are normal.
For more on Normality and the Fat Pencil
http://www.statit.com/support/quality_practice_tips/normal_probability_plot_
interpre.shtml
Walter Shewhart
Developer of Control Charts in the late 1920’s
You did Control Charts in DFM. There the emphasis was on tolerances. Here the
emphasis is on determining if a process is in control. If the process is in control, we want
to know the capability.
www.york.ac.uk/.../ histstat/people/welcome.htm
What does the data tell us about our
process?
SPC is a continuous improvement tool which minimizes tampering or
unnecessary adjustments (which increase variability) by distinguishing
between special cause and common cause sources of variation
Control Charts have two basic uses:
Give evidence whether a process is operating in a state of statistical
control and to highlight the presence of special causes of variation so
that corrective action can take place.
Maintain the state of statistical control by extending the statistical
limits as a basis for real time decisions.
If a process is in a state of statistical control, then capability studies my be
undertaken. (But not before!! If a process is not in a state of statistical
control, you must bring it under control.)
SPC applies to design activities in that we use data from manufacturing to
predict the capability of a manufacturing system. Knowing the
capability of the manufacturing system plays a crucial role in selecting
the concepts.
Voice of the Process
Control limits are not spec limits.
Control limits define the amount of fluctuation that a
process with only common cause variation will have.
Control limits are calculated from the process data.
Any fluctuations within the limits are simply due to
the common cause variation of the process.
Anything outside of the limits would indicate a
special cause (or change) in the process has occurred.
Control limits are the voice of the process.
The capability index depends on the spec limit and
the process standard deviation.
Cp = (allowable range)/6s = (USL - LSL)/6s
LSL
USL (Upper Specification Limit)
LCL
UCL (Upper Control Limit)
http://lorien.ncl.ac.uk/ming/spc/spc9.htm
Upper Control
Limit for 2008
Lower Control
Limit for 2008
Minitab prints results in the Session window that lists any
failures.
Test Results for I Chart of StackedTotals by C4
TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 129
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 15, 110, 111, 112, 113
TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on
one side of CL).
Test Failed at points: 52, 66, 119, 160, 161
TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on
one side of CL).
Test Failed at points: 91, 97
TEST 7. 15 points within 1 standard deviation of center line (above and below CL).
Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200
This chart is extremely helpful for deciding what
statistical technique to use.
X Data
Multiple Xs
Single X
One-sample ttest
Two-sample ttest
ANOVA
Simple
Linear
Regression
Discrete
Logistic
Regression
Y Data
Chi-Square
Discrete X Data Continuous
Continuous
Continuous
Y Data
Discrete
Single Y
Multiple Ys
Y Data
Discrete X Data Continuous
Multiple
Logistic
Regression
Multiple
Logistic
Regression
ANOVA
Multiple
Linear
Regression
When to use ANOVA
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The use of ANOVA is appropriate when
 Dependent variable is continuous
 Independent variable is discrete, i.e. categorical
 Independent variable has 2 or more levels under study
 Interested in the mean value
 There is one independent variable or more
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We will first consider just one independent variable
ANOVA Analysis of Variance
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Used to determine the effects of categorical independent
variables on the average response of a continuous variable
Choices in MINITAB
 One-way ANOVA
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Two-way ANOVA
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Use with two factors, varied over multiple levels
Balanced ANOVA
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Use with one factor, varied over multiple levels
Use with two or more factors and equal sample sizes in each cell
General Linear Model
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Use anytime!
Practical Applications
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Determine if our break pedal sticks more than other
companies
Compare 3 different suppliers of the same
component
Compare 6 combustion recipes through simulation
Determine the variation in the crush force
Compare 3 distributions of M&M’s
And MANY more …
The null hypothesis for ANOVA is that there is no
difference between years.
General Linear Model: StackedTotals versus C4
Factor Type Levels Values
C4
fixed
3 2008, 2010, 2011
This p value indicates
Analysis of Variance for StackedTotals, using Adjusted SS for Tests
that the assumption that
there is no difference
Source DF
Seq SS Adj SS Adj MS F
P
between years is not
C4
2
6.6747 6.6747 3.3374 4.71 0.010
correct!
Error 203
143.8559 143.8559 0.7086
Total 205
150.5306
S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%
What are some conclusions that you can reach?
Is there a statistical difference between years?
Main Effects Plot for StackedTotals
Fitted Means
7.9
Mean
7.8
7.7
7.6
7.5
7.4
2008
2010
Year
2011
The p value indicates that there is a difference between
the years. The Tukey printout tells us which years are
different.
Grouping Information Using Tukey Method and 95.0% Confidence
C4
2010
2008
2011
N
57
86
63
Mean
7.9
7.7
7.4
Grouping
A
AB
B
Means that do not share a letter are significantly different.
The averages for 2010 and 2008 are not statistically different. The averages for
2008 and 2011 are not statistically different.
Command:
>Stat>Basic Statistics>Display Descriptive Statistics
Why would we care about this data in design?
If I am a plant manager, do I like one distribution better
than another?
This is a density description of the data.
>Stat>Basic Statistics>Normality Test
Select 2008
The Anderson-Darling normality test is used to
determine if data follow a normal distribution.
If the p-value is lower than the predetermined level of significance, the data
do not follow a normal distribution.
Command:
>Stat>Control Charts>Variable Charts for Individuals>Individuals
When doing control charts for ME470, select all tests.
It may be hard to see, but highlight the “tests” tab.
Minitab prints results in the Session window that lists any
failures.
Test Results for I Chart of StackedTotals by C4
TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 129
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 15, 110, 111, 112, 113
TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on
one side of CL).
Test Failed at points: 52, 66, 119, 160, 161
TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on
one side of CL).
Test Failed at points: 91, 97
TEST 7. 15 points within 1 standard deviation of center line (above and below CL).
Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200
Upper Control
Limit for 2008
Lower Control
Limit for 2008
Command:
>Stat>ANOVA>General Linear Model
What are some conclusions that you can reach?
Is there a statistical difference between years?
Main Effects Plot for StackedTotals
Fitted Means
7.9
Mean
7.8
7.7
7.6
7.5
7.4
2008
2010
Year
2011
The null hypothesis for ANOVA is that there is no
difference between years.
General Linear Model: StackedTotals versus C4
Factor Type Levels Values
C4
fixed
3 2008, 2010, 2011
This p value indicates
Analysis of Variance for StackedTotals, using Adjusted SS for Tests
that the assumption that
there is no difference
Source DF
Seq SS Adj SS Adj MS F
P
between years is not
C4
2
6.6747 6.6747 3.3374 4.71 0.010
correct!
Error 203
143.8559 143.8559 0.7086
Total 205
150.5306
S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%
Command:
>Stat>ANOVA>General Linear Model
The p value indicates that there is a difference between
the years. The Tukey printout tells us which years are
different.
Grouping Information Using Tukey Method and 95.0% Confidence
C4
2010
2008
2011
N
57
86
63
Mean
7.9
7.7
7.4
Grouping
A
AB
B
Means that do not share a letter are significantly different.
The averages for 2010 and 2008 are not statistically different. The averages for
2008 and 2011 are not statistically different.
Here is a useful reference if you feel that you need
to do more reading.
http://www.StatisticalPractice.com
This recommendation is thanks to Dr. DeVasher.
You can also use the help in Minitab for more information.
Let’s look at what happened with
plain M&M’s
What do you see with the
boxplot?
Do we see anything that looks unusual?
General Linear Model: stackedTotal versus StackedYear
Factor
StackedYear
Type Levels
fixed
4
Values
2004, 2005, 2006, 2009
Analysis of Variance for stackedTotal, using Adjusted SS for Tests
Source
DF
Seq SS Adj SS Adj MS
F
P
StackedYear 3 1165.33 1165.33 388.44 149.39 0.000 Look at low P-value!
Error
266 691.63 691.63 2.60
Total
269 1856.96
S = 1.61249 R-Sq = 62.75% R-Sq(adj) = 62.33%
Unusual Observations for stackedTotal
Obs stackedTotal Fit
SE Fit Residual
25
27.0000 23.4667 0.2082 3.5333
34
20.0000 23.4667 0.2082 -3.4667
209
40.0000 21.7917 0.1700 18.2083
215
21.0000 17.4917 0.2082 3.5083
St Resid
2.21 R
-2.17 R
11.36 R
2.19 R
R denotes an observation with a large standardized residual.
Grouping Information Using Tukey Method and 95.0% Confidence
StackedYear N Mean Grouping
2004
60 23.5 A
2006
90 21.8 B
2005
60 20.7 C
2009
60 17.5
D
Means that do not share a letter are significantly different.
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable stackedTotal
All Pairwise Comparisons among Levels of StackedYear
StackedYear = 2004 subtracted from:
StackedYear
2005
2006
2009
Lower Center
-3.531 -2.775
-2.365 -1.675
-6.731 -5.975
Upper -------+---------+---------+---------2.019
(---*---)
-0.985
(-*--)
-5.219 (--*--)
-------+---------+---------+---------5.0
-2.5
0.0
Zero is not contained in the intervals. Each year is
statistically different. (2004 got the most!)
StackedYear = 2005 subtracted from:
StackedYear Lower Center Upper -------+---------+---------+--------2006
0.410 1.100 1.790
(-*--)
2009
-3.956 -3.200 -2.444
(--*--)
-------+---------+---------+---------5.0
-2.5
0.0
StackedYear = 2006 subtracted from:
StackedYear Lower Center Upper -------+---------+---------+--------2009
-4.990 -4.300 -3.610
(--*--)
-------+---------+---------+---------5.0
-2.5
0.0
Implications for design
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Is there a difference in production
performance between the plain and peanut
M&Ms?
Individual Quiz
Name:____________
Section No:__________
CM:_______
You will be given a bag of M&M’s. Do NOT eat the M&M’s.
Count the number of M&M’s in your bag. Record the number of each color,
and the overall total. You may approximate if you get a piece of an M&M.
When finished, you may eat the M&M’s. Note: You are not required to eat the
M&M’s.
Color
Brown
Yellow
Red
Orange
Green
Blue
Other
Total
Number
%
Instructions for Minitab Installation
Minitab on DFS: