The purpose of the one-way analysis of variance (ANOVA) is to determine whether three or more groups have the same mean.

**THE PURPOSE OF ONE-WAY ANALYSIS OF VARIANCE**

The purpose of the one-way analysis of variance
(ANOVA) is to determine whether three or more groups have the same mean (i.e., *H*_{0}: *μ*_{1} = *μ*_{2} = *μ*_{3}, . . . , *μ** _{k}*). It is a generalization of the

The analysis of variance is based on a linear model
that says that the response for group *j*,
denoted *X _{j}*, satisfies
Equation 13.1 for a one-way ANOVA:

*X*_{ij} = μ_{j} + ε_{ij} (13.1)

where *i*
is the ith observation from the *j*th
group *j* = 1, 2, . . . , *k*; *j*
is the group label and we have *k* ≥ 3 groups; *m** _{j}* is the mean for group

The test statistic is the ratio of estimates of two
sources of variation called the within-group variance and the between-group
variance. If the treatment makes a difference, then we expect that the
between-group variance will exceed the within-group variance. These variances
or sums of squares when normalized have independent chi-square distributions
with *n _{w}* and

The ratio of these mean squares is the test
statistic for the analysis of variance. When the means are equal, this ratio
has an *F* distribution with *n _{b}* degrees of freedom in the
numerator and

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