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# 2.11: Analysis of Variance and the F-Distribution

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## The F-Distribution and Testing Two Variances

Project: The History of Statistics

Some students can happily work in the abstract world of numbers and symbols indefinitely, but most need and appreciate seeing the human side of statistics along with its theory and applications. Statistics has a rich history with many interesting characters for students to explore. Linking a statistical method to a memorable story or person will improve the student’s ability to retain what they have learned in the course.

Objective: Find and report on the history of the development of the $t$ distribution and the $F$ distribution.

Procedure: Answer the following questions for each distribution in a written report and/or a class presentation.

• When was the distribution first used?
• Who discovered the distribution?
• What motivated the distributions discovery?
• How did the distribution get its name?
• What other notable work was taking place in science and mathematics at the time the distribution was first used?
• What was happening historically, politically, and in the art world at the time the distribution was first used?
• What developments have been made concerning the distribution since the time of its first use and who made these developments?
• What new uses have been found for the distribution since the time of its first use and who found these applications?

## The One-Way ANOVA Test

Explore: Assessing Variance with the F-test and ANOVA When Data is Approximately Normally Distributed

It is sometimes difficult for students to remember all the requirements and sensitivities of a particular test. Students have learned two tests that compare the variances of two sets of data. The $F-$test is only accurate when the data is normally distributed, but the ANOVA test is not as sensitive to small deviations from normality. By applying both of these tests to data that meets the normal requirement and to data that does not meet the strict normal requirement, students will be able to experience the magnitude of the discrepancy in the results. The experience will help them to remember the normal requirement for the $F-$test.

Objective: To compare the results of the $F-$test and the ANOVA test when analyzing the variance of data that is normally distributed and when analyzing data that is approximately normally distributed.

Procedure:

Part One: Normally Distributed Data

1. Take two samples from a population you know to be normally distributed. For example, you can use the heights of women and the heights of men since you know height to be normally distributed.

2. Perform an $F-$test on the variances of the two data sets.

3. Perform an ANOVA test on the variances of the two data sets.

4. Compare the results.

Part Two: Data that is approximately normally distributed.

5. Take two samples from a population that you know has an approximate normal distribution.

6. Perform an $F-$test on the variances of the two data sets.

7. Perform an ANOVA test on the variances of the two data sets.

8. Compare the results.

Analysis:

(9) Analyze the normality requirement of the $F-$test using what you have learned from the tests just conducted.

## The Two-Way ANOVA Test

Project: Technology Handbook

Throughout the course, students have used technology to perform statistical calculations. Excel and the TI-84 calculator have been used most frequently. In this assignment, student will gather and solidify their knowledge for their own future use and for the use of other students.

Objective: To write a clear, precise handbook that could be used by a person with no prior knowledge of the technology to perform statistical calculations with Excel and with the TI-84 calculator.

Procedure:

1. Make an index.

• Review what you have done with technology in this class and decide what should be included in your handbook.
• Find a user friendly way to organize the information. Do you want to have an Excel section and a TI-84 section, or do you want to categorize by statistical topic? Maybe you have a different method to organize the material.

2. Write clear, concise directions so that each statistical calculation can be preformed with each technological tool.

• Include pictures.
• Ask someone who is not knowledgeable in this area to read over your work and identify areas that are unclear.

This project can be done in conjunction with the English department. Technical writing of this sort is a valuable skill for students to develop.

Feb 23, 2012

Aug 19, 2014