Chap 12-41
Within-Group Variation
Summing the variation
within each group and then
adding over all groups
i
k
n
SSW
MSW
T
Mean Square Within =
SSW/degrees of freedom
2
1
1
)
x
x
(
SSW
i
ij
n
j
k
i
j
Chap 12-42
Within-Group Variation
(continued)
Group 1
Group 2
Group 3
Response, X
2
2
2
12
2
1
11
)
x
x
(
...
)
x
x
(
)
x
x
(
SSW
k
kn
k
3
x
1
x
2
x
Chap 12-43
One-Way ANOVA Table
Source of
Variation
df
SS
MS
Between
Samples
SSB
MSB =
Within
Samples
n
T
- k
SSW
MSW =
Total
n
T
- 1
SST =
SSB+SSW
k - 1
MSB
MSW
F ratio
k
= number of populations
n
T
= sum of the sample sizes from all populations
df = degrees of freedom
SSB
k - 1
SSW
n
T
- k
F =
Chap 12-44
One-Factor ANOVA
F Test Statistic
Test statistic
MSB
is mean squares
between
variances
MSW
is mean squares
within
variances
Degrees of freedom
df
1
= k
–
1
(k = number of populations)
df
2
= n
T
–
k
(n
T
= sum of sample sizes from all populations)
MSW
MSB
F
H
0
:
μ
1
=
μ
2
= …
=
μ
k
H
A
: At least two population means are different
Chap 12-45
Interpreting One-Factor ANOVA
F Statistic
The F statistic is the ratio of the
between
estimate of variance and the
within
estimate
of variance
The ratio must always be positive
df
1
= k -1
will typically be small
df
2
= n
T
- k
will typically be large
The ratio should be
close to 1
if
H
0
:
μ
1
=
μ
2
= … = μ
k
is
true
The ratio will be
larger than 1
if
H
0
:
μ
1
=
μ
2
= … = μ
k
is
false
Chap 12-46
Table 14.5
(p. 544)
Percentage Points of (
F
(v
1
, v
2
)
Distributions
= 5
