The Distribution
of Fratios

The FRatio:
The heart of ANOVA is analyzing the total variability into
these two components, the mean square between and mean square
within.
Once we have analyzed the total variability into its two
basic components we simply compare them. The comparison
is made by computing the Fratio. For independentmeasures
ANOVA the Fratio has the following structure:
or, using the vocabulary of ANOVA,
Characteristics of the Fratio
 The numerator and denominator of the ratio measure exactly
the same variance when the null hypothesis is true. Thus:
when Ho is true, F is about 1.00.
 Fratios are always positive, because the Fratio is
a ratio of two variances, and variances are always positive.
Given these two factors, we can sketch the distribution of
Fratios. The distribution piles up around 1.00, cuts off
at zero, and tapers off to the right.
Degrees of Freedom:
Note that the exact shape depends on the degrees of freedom
of the two variances. We have two separate degrees of freedom,
one for the numerator (sum of squares between) and the other
for the denominator (sum of squares within). They depend on
the number of groups and the total number of observations.
The exact number of degrees of freedom follows these two formulas
(k is the number of groups, N is the total number of observations):
Two F Distributions:
Here are two examples of F distributions. They differ in the
degrees of freedom:
 For the data about learning under different termperature
condtions (discussed above), the df(between)=31=2, and
the df(within)=153=12. We can look up the critical value
of F (.05) and find that it is 3.88. The observed F=11.28,
so we reject the null hypothesis. The Fratio distribution
is:
 For data where df=5,30 (6 groups, 36 observations),
the Fratio distribution is:
