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Statistics and ANOVA
ME 470
Fall 2009
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.
How Do We Describe the World?
Quiz for the day
 You need to install Minitab on your
computers. Sign on as localmgr
 >Start>Run
 \\tibia\Public\Course Software\Minitab
 Double click on Minitab R15 Install
 What can we say about our M&Ms?

>Stat>Basic Statistics>Display Descriptive Statistics
2004, 2005, 2006 Data
Descriptive Statistics: stackedTotal
Variable
StackedYear
stackedTotal
2004
2005
2006
N
60
60
90
N*
0
0
0
Variable
StackedYear
stackedTotal
2004
2005
2006
Q1
23.000
20.000
21.000
Mean SE Mean StDev
23.467 0.188 1.455
20.692 0.135 1.046
21.792 0.232 2.202
Median Q3
23.500 24.000
21.000 21.000
22.000 22.000
Minimum
20.000
18.000
19.000
Maximum
27.000
23.000
40.000
Why would we care about this in design?
Assessing Shape: Boxplot
largest value excluding outliers
Boxplot of BSNOx
2.45
Q3
2.40
(Q2), median
BSNOx
2.35
2.30
Q1
2.25
http://en.wikipedia.org/wiki/B
ox_plot
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.
>Stat>Basic Statistics>Normality Test
Select StackedTotal_2004
Anderson-Darling normality test:
Used to determine if data follow a normal distribution. If the p-value is lower than the
pre-determined 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.
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.
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 this 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 is defined as:
Cp = (allowable range)/6s = (USL - LSL)/6s
LSL
LCL
USL (Upper Specification Limit)
UCL (Upper Control Limit)
http://lorien.ncl.ac.uk/ming/spc/spc9.htm
>Stat>Control Charts>Variable Charts for Individuals>I-MR
Upper Control
Limit
Lower Control
Limit
Absolute difference between two adjacent points.
Are the 3 Distributions Different?
X Data
Single X
Multiple Xs
Y Data
Logistic
Regression
One-sample ttest
Two-sample ttest
Simple
Linear
Regression
Discrete
Discrete X Data Continuous
Continuous
Continuous
Y Data
Discrete
Single Y
Chi-Square
ANOVA
Multiple Ys
Y Data
Discrete X Data Continuous
Multiple
Logistic
Regression
Multiple
Logistic
Regression
ANOVA
Multiple
Linear
Regression
When to use ANOVA

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

We will first consider just one independent variable
Practical Applications






Compare 3 different suppliers of the same
component
Compare 4 test cells
Compare 2 performance calibrations
Compare 6 combustion recipes through simulation
Compare 3 distributions of M&M’s
And MANY more …
ANOVA Analysis of Variance


Used to determine the effects of categorical independent
variables on the average response of a continuous variable
Choices in MINITAB
 One-way ANOVA


Two-way ANOVA


Use with two factors, varied over multiple levels
Balanced ANOVA


Use with one factor, varied over multiple levels
Use with two or more factors and equal sample sizes in each cell
General Linear Model

Use anytime!
>Stat>ANOVA>General Linear Model
15
25
Effect of Year on M&M Production
General Linear Model: stackedTotal versus StackedYear
Factor
StackedYear
Type Levels
fixed
3
Values
2004, 2005, 2006
Analysis of Variance for stackedTotal, using Adjusted SS for Tests
Source
StackedYear
Error
Total
DF Seq SS Adj SS Adj MS F P
2 235.27 235.27 117.63 39.22 0.000
207 620.89 620.89 3.00
209 856.16
S = 1.73189 R-Sq = 27.48% R-Sq(adj) = 26.78%
Unusual Observations for stackedTotal
Obs stackedTotal Fit SE Fit Residual St Resid
25
27.0000 23.4667 0.2236 3.5333 2.06 R
34
20.0000 23.4667 0.2236 -3.4667 -2.02 R
209
40.0000 21.7917 0.1826 18.2083 10.57 R
R denotes an observation with a large standardized residual.
This low p-value
indicates that at least
one year is different
from the others.
>Stat>ANOVA>General Linear Model
We use the Tukey comparison to determine if the years are different.
Confidence intervals that contain zero suggest no difference.
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable stackedTotal
All Pairwise Comparisons among Levels of StackedYear
StackedYear = 2004 subtracted from:
StackedYear
2005
2006
Lower Center Upper ---+---------+---------+---------+---3.522 -2.775 -2.028 (---*----)
-2.357 -1.675 -0.993
(----*---)
---+---------+---------+---------+---3.0
-1.5
0.0
1.5
Because “0.0” is not contained in the range, we concluded that 2004 is statistically
different from both 2005 and 2006.
StackedYear = 2005 subtracted from:
StackedYear
2006
Difference
of Means
1.100
SE of
Adjusted
Difference T-Value P-Value
0.2886
3.811 0.0005
StackedYear = 2005 subtracted from:
StackedYear
2006
Lower Center Upper ---+---------+---------+---------+--0.4183 1.100 1.782
(---*----)
---+---------+---------+---------+---3.0
-1.5
0.0
1.5
Again, because “0.0” is not in the range, we conclude that 2005 is statistically
different than 2006.
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
%