ANOVA is an acronym for "analysis of variance", it is a test for equality of
multiple means.
One-way ANOVA or single factor analysis of variance means that we are testing
randomly selected, independent samples of normal populations with equal variances
to that have one distinguishing factor, treatment, or characteristic.
We assume that the factor does not alter the means of the respective populations
and we test to see if there is significant evidence that one of the means is
different.
The distribution for this test is called an \(F\)-distribution and
\(F\sim F_{df(num),df(denom)}\) where \(df(num)\) and \(df(denom)\) are two separate degrees
of freedom.
For the one-way ANOVA test with \(k\) groups and \(n\) total values,
\(df(num)=k-1\) and \(df(denom)=n-k\).
13.2 The F Distribution and the F-Ratio
recommended:
9-23, 61-63
Assume that we have collected \(k\) groups of data. Each group may have a
separate number \(n_{1}, n_{2}, \dots,n_{k}\) of data points.
We can calculate the means and standard deviations of these samples, but we are
wondering if we can show evidence that these means are different.
This knowledge would be particularly useful if we were comparing similar
populations that were assigned a single different medical treatment, for example.
13.3 Facts About the F Distribution
recommended:
24-42, 64-74
The mean for \(F\sim F_{df(num),df(denom)}\) where
\(\displaystyle{\mu=\frac{df(denom)}{df(denom)-2}}\)
\(F\)-distributions are skewed right.
As the degrees of freedom for the numerator and denominator get larger, the
distribution approximates a normal distribution.
13.4 Test of Two Variances
recommended:
43-58, 75-83
In the test of two population variances, our null hypothesis is
\(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) and our test statistic is
\(\displaystyle{F=s_{1}^{2}/s_{2}^{2}}\)
For the test of two variances with two groups of \(n_{1}\) and \(n_{2}\) values,
respectively, \(df(num)=n_{1}-1\) and \(df(denom)=n_{2}-1\).
A test of two variances may be left-tailed, right-tailed, or two-tailed.