The distribution used for the hypothesis test is a new one. It is called the F distribution,
named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a
fraction). There are two sets of degrees of freedom; one for the numerator and one for
the denominator.
For example, if
F
F follows an
F
F distribution and the degrees of freedom for the
numerator are 4 and the degrees of freedom for the denominator are 10, then
FF ~
F
4
,
10
F
4
,
10
.
To calculate the
F
F
ratio, two estimates of the variance are made.
- Variance between samples: An estimate of
σ
2
σ
2
that is the variance of the sample
means. If the samples are different sizes, the variance between samples is weighted to
account for the different sample sizes. It is also called variation due to treatment or
explained
variation.
- Variance within samples: An estimate of
σ
2
σ
2
that is the average of the sample
variances (also known as a pooled variance). When the sample sizes are different, the
variance within samples is weighted. It is also called the variation due to error or
unexplained variation.
SS
between
=
SS
between
= the sum of squares that represents the variation among the different
samples.
SS
within
=
SS
within
= the sum of squares that represents the variation within samples that is
due to chance.
To find a "sum of squares" means to add together squared quantities which, in some
cases, may be weighted. We used sum of squares to calculate the sample variance and
the sample standard deviation in Chapter 2. In this very brief overview of ANOVA, we
will not go into detail explaining how
SS
between
SS
between
and
SS
within
SS
within
are calculated. If you take
another statistics course, you will cover sum of squares in detail. Your calculator can
easily do the calculations of
SS
between
SS
between
and
SS
within
SS
within
. Remember that these two quantities
are measures of variability or variation.
df
between
=
k
-
1
df
between
=k-1 where
k
k is the number of groups (samples).
This is the degrees of freedom for the numerator.
df
within
=
N
-
k
df
within
=N-k where
N
N is the total sample size.
This is the degrees of freedom for the denominator.
MSMS means "mean square."
MS
between
MS
between
is the variance between groups and
MS
within
MS
within
is the variance within groups.
MS
between
=
SS
between
df
between
=
SS
between
k
−
1
MS
between
=
SS
between
df
between
=
SS
between
k
−
1
MS
within
=
SS
within
df
within
=
SS
within
N
−
k
MS
within
=
SS
within
df
within
=
SS
within
N
−
k
The ANOVA test depends on the fact that
MS
between
MS
between
can be influenced by population
differences among means of the several groups. Since
MS
within
MS
within
compares values of
each group to its own group mean, the fact that group means might be different does
not affect
MS
within
MS
within
.
The null hypothesis says that all groups are samples from populations having the same
normal distribution. The alternate hypothesis says that at least two of the sample
groups come from populations with different normal distributions. If the null hypothesis
is true,
MS
between
MS
between
and
MS
within
MS
within
should both estimate the same value.
The F-statistic or F-ratio, as it is often called, is
F
=
MS
between
MS
within
F=
MS
between
MS
within
If
MS
between
MS
between
and
MS
within
MS
within
estimate the same value (following the belief that
H
o
H
o
is
true), then the F-ratio should be approximately equal to 1. Only sampling errors
would contribute to variations away from 1. As it turns out,
MS
between
MS
between
consists of
the population variance plus a variance produced from the differences between the
samples.
MS
within
MS
within
is an estimate of the population variance. Since variances are
always positive, if the null hypothesis is false,
MS
between
MS
between
will be larger than
MS
within
MS
within
.
The F-ratio will be larger than 1.
The ANOVA hypothesis test is always right-tailed because larger F-values are
way out in the right tail of the F-distribution curve and tend to make us reject
H
o
H
o
.
The notation for the F distribution is
FF ~
F
df(num)
,
df(denom)
F
df(num)
,
df(denom)
where
df(num)
=
df
between
df(num)=
df
between
and
df(denom)
=
df
within
df(denom) =
df
within
The mean for the F distribution is
m
=
df(num)
df(denom)
−
1
m=
df(num)
df(denom)
−
1
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