(ANalysis Of VAriance)
Daniel Heaton
MBA 634
March 27, 2006
What Will Be Covered
• What ANOVA is and where it comes from
• How ANOVA can be used in Quality
Management
• The basic parts of ANOVA
• How ANOVA works and how it can be
performed using Excel
• Example and Exercise for ANOVA
Application
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What ANOVA is
• An ANOVA is a guide for determining whether or
not an event was most likely due to the random
chance of natural variation.
• Or, conversely, the same method provides
guidance in saying with a specific level of
confidence that a certain factor (X) or factors (X,
Y, and/or Z) were the more likely reason for the
event.
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Where ANOVA Comes From
Ronald Aylmer Fisher (1890-1962)
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Brainstorming Exercise
• Why would you want to know if the
difference between data sets is statistically
significant?
• What kinds of data are used or collected in
your organization that ANOVA would be
useful for?
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The Different Types of ANOVA
One-way between groups
• You are looking at the differences between the
groups.
• There is only one factor (or result) which you are
using to define the groups.
• This is the simplest version of ANOVA.
• This type of ANOVA can also be used to compare
variables between different groups.
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The Different Types of ANOVA
One-way repeated measures
• A one way repeated measures ANOVA is used
when you have a single group on which you have
measured something a few times.
• You would use a one-way repeated measures
ANOVA to see if results changed significantly
over time.
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The Different Types of ANOVA
Two-way between groups
• A two-way between groups ANOVA is used to
look at complex groupings.
• Examines the effects of two different factors and
their interactions.
• Each of the main effects are one-way tests. The
interaction effect is simply asking "is there any
significant difference in performance when you
consider two factors acting together".
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The Different Types of ANOVA
Two-way repeated measures
• This version of ANOVA simple uses the repeated
measures structure of the “One-way repeated
measures” method and includes the interaction
effect of the “Two-way between groups” method.
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The ANOVA Table
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The Basic Parts of ANOVA
SS or Sum of Squares
• This is the measure of the variation around the
mean. There are usually three different values of
SS calculated:
– SSG measures variation of the group means around the
overall mean (Between Groups)
– SSE measures the variation of each observation around
its group mean (Within Groups)
– SST measures variation of the data around the overall
mean (Total)
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The Basic Parts of ANOVA
df or Degrees of Freedom
• This is the factor that adjusts for how large the
groups are and the number of groups being
considered. They are calculated as follows:
– Number of Groups (j) – 1 for SSG
– Sample Size (n) – Number of Groups (j) for SSE
– Sample Size (n) - 1 for SST
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The Basic Parts of ANOVA
MS = Mean Square = SS/df
• This is like a standard deviation. Its numerator is
the sum of squared deviations (SS), divided by the
appropriate number of degrees of freedom.
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The Basic Parts of ANOVA
F (F-Statistic or F-Ratio) = MSG/MSE
• This tells you the proportion of variation between
the groups compared to the variation within the
groups.
– In general, the larger this value is, the more likely the
variation between the groups is significant.
– The level of significance is determined by comparing it
to the F-Critical value for the samples. If the F-Statistic
is larger than F-Critical, then the variation between the
groups is statistically significant.
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How to Perform ANOVA Using Excel
1. Enter the data into Excel
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How to Perform ANOVA Using Excel
1. Enter the data into excel
2. Access ANOVA function using: Tools >
Data Analysis > ANOVA: Single Factor
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How to Perform ANOVA Using Excel
1. Enter the data into excel
2. Access ANOVA function using: Tools >
Data Analysis > ANOVA: Single Factor
3. Select Range of Cells where data is
located for “Input Range” and select other
options as appropriate and click “OK”
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How to Perform ANOVA Using Excel
Excel Output
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A Real World Example
• Three inspectors wanted to see how
accurately they measure a dimension of
parts they inspect.
• They each measured the same 10 parts in
random order multiple times and the
measurements are recorded as follows.
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A Real World Example
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A Real World Example
• Since they want to look at different parts
and different operators with repeated
measurements, they use the “Anova: TwoFactor With Replication” function.
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A Real World Example
Excel Results
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A Real World Example
Excel Results
• The variation within the inspectors measurements,
between the inspectors, and of interactions
between the two were all not significant.
• However, this data helps them to see that most of
the variation is due to the gage and not the
operator.
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ANOVA Exercise
• Three friends want to see who is the best
bowler.
• They each play a different number of games
and record their scores.
• Analyze the data and determine which one
is the best bowler, and if the results are
significant or by chance.
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ANOVA Exercise
Enter the following data into Excel and
analyze it using the “Anova: Single Factor”
function.
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ANOVA Exercise
Procedure
• Select: Tools > Data
Analysis > Anova:
Single Factor
• Select Input Range
and other values as
shown.
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ANOVA Exercise
Results
• Joe has the highest
average score
• The results are
statistically
significant and not
due to chance
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Summary
• ANOVA is a useful and powerful tool for
determining if differences are statistically
significant.
• It can also be used to establish cause and effect
relationships with a specific degree of certainty
• Excel has three ANOVA functions that can be used
fro basic analysis of variance.
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Readings List
• The basic concepts of ANOVA can be found
in almost any statistics text book.
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Readings List
• Books
– Damon, Richard A., Experimental design, ANOVA, and
regression, New York, Harper & Row, 1987.
– Miller, Rupert G., Beyond ANOVA, basics of applied
statistics, New York, Wiley, 1986.
– Rutherford, Andrew, Introducing Anova and Ancova :
a GLM approach, Thousand Oaks, SAGE, 2001.
– Weiss, David J., Analysis of variance and functional
measurement : a practical guide, New York, Oxford
University Press, 2006.
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Readings List
• Online Resources
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http://www.isixsigma.com/library/content/c021111a.asp
http://www.uwsp.edu/psych/cw/statmanual/anovaterms.html
http://www.csse.monash.edu.au/~smarkham/resources/anova.htm
www.math.yorku.ca/SCS/StatResource.html
www.statsci.org/teaching.html
http://www.sportsci.org/resource/stats/index.html
http://www.uwsp.edu/psych/cw/statmanual/index.html
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