(ANalysis Of VAriance)

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(ANalysis Of VAriance). ANOVA. 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

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(ANalysis Of VAriance)

ANOVA

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

MBA 634

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.

MBA 634

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.

MBA 634

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.

MBA 634

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".

MBA 634

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 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)

MBA 634

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
• Enter the data into Excel

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How to Perform ANOVA Using Excel
• Enter the data into excel
• Access ANOVA function using: Tools > Data Analysis > ANOVA: Single Factor

MBA 634

How to Perform ANOVA Using Excel
• Enter the data into excel
• Access ANOVA function using: Tools > Data Analysis > ANOVA: Single Factor
• Select Range of Cells where data is located for “Input Range” and select other options as appropriate and click “OK”

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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
• Since they want to look at different parts and different operators with repeated measurements, they use the “Anova: Two-Factor 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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• The basic concepts of ANOVA can be found in almost any statistics text book.

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• 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.

MBA 634