F Distribution and One-Way ANOVA
One-way analysis of variance, or ANOVA, tests whether the means of several groups differ. This chapter introduces the F distribution and the F-ratio on which the test is based, explains how one-way ANOVA compares variation between groups with variation within groups, and describes the main properties of the F distribution.
The F Distribution and the F-Ratio
The F distribution is a family of curves used when comparing variances or the means of several groups. It is right-skewed, never negative, and its exact shape depends on two separate degrees-of-freedom values. The test statistic in these procedures is the F-ratio, formed by dividing one estimate of variance by another. When the F-ratio is close to one the estimates agree, and a large F-ratio signals that the quantities being compared differ more than chance alone would explain.
One-Way ANOVA
One-way ANOVA tests whether three or more group means are equal, using a single factor to define the groups. The null hypothesis is that all the population means are the same, while the alternative is that at least one differs. The method compares the variation between the group means with the variation within the groups: if the between-group variation is large relative to the within-group variation, the F-ratio is large and the null hypothesis of equal means is rejected. Several conditions, including roughly equal variances, are assumed.
Facts About the F Distribution
The F distribution has several defining features. Its values are always positive because it is built from ratios of variances, and the curve is skewed to the right rather than symmetric. Its shape is governed by the degrees of freedom of the numerator and the denominator, and as both grow the distribution becomes more symmetric. Besides one-way ANOVA, the F distribution is also used to test whether two populations have equal variances, its second common application in this chapter.