# Chapter 11: Analysis of Variance and the F-Distribution

**Advanced**Created by: CK-12

## Introduction

In previous chapters, we learned how to conduct hypothesis tests that examined the relationship between two variables. Most of these tests simply evaluated the relationship of the means of two variables. However, sometimes we also want to test the variance, or the degree to which observations are spread out within a distribution. In the figure below, we see three samples with identical means (the samples in red, green, and blue) but with very different variances:

So why would we want to conduct a hypothesis test on variance? Let’s consider an example. Suppose a teacher wants to examine the effectiveness of two reading programs. She randomly assigns her students into two groups, uses a different reading program with each group, and gives her students an achievement test. In deciding which reading program is more effective, it would be helpful to not only look at the mean scores of each of the groups, but also the “spreading out” of the achievement scores. To test hypotheses about variance, we use a statistical tool called the \begin{align*}F\end{align*}-distribution.

In this lesson, we will examine the difference between the \begin{align*}F\end{align*}-distribution and Student’s \begin{align*}t\end{align*}-distribution, calculate a test statistic with the \begin{align*}F\end{align*}-distribution, and test hypotheses about multiple population variances. In addition, we will look a bit more closely at the limitations of this test.

### Chapter Summary

## Summary

This chapter expands upon the previous lesson’s introduction to variance, focusing on examining the f-max test and one- and two-way ANOVA tests.

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