Analysis of
Variance
Analysis of Variance
 Single
classification analysis of variance
determines whether a relationship exists
between a dependent variable and
several classifications of one independent
variable.
 Multiple classification analysis of variance
determines the relationship between one
dependent variable and classifications of
two or more independent variables.
Analysis of Variance
 Since
the variance (or its square
root, the standard deviation) is an
average distance of the raw scores
from the mean of that distribution,
this functional relationship can be
used to determine mean differences
by analyzing variances.
Analysis of Variance
x1
1
2
3
4
5
15
x1 = 3
x2
3
4
5
6
7
25
x2 = 5
x3
5
6
7
8
9
35
x3 = 7
Within Group Variation
x1
1
2
3
4
5
15
x1 = 3
x-x
(x - x)
-2
-1
0
1
2
0
4
1
0
1
4
10
2
Within Group Variation
x2
3
4
5
6
7
25
x2 = 5
x-x
(x - x)
-2
-1
0
1
2
0
4
1
0
1
4
10
2
Within Group Variation
x3
5
6
7
8
9
35
x3 = 7
x-x
(x - x)
-2
-1
0
1
2
0
4
1
0
1
4
10
2
SS Within = 30
Total Variation
x1
1
2
3
4
5
x-x
(x - x)
-4
-3
-2
-1
0
16
9
4
1
0
30
2
Total Variation
x2
3
4
5
6
7
x-x
(x - x)
-2
-1
0
1
2
4
1
0
1
4
10
2
Total Variation
x3
5
6
7
8
9
SS Total = 70
x-x
0
1
2
3
4
(x - x)
0
1
4
9
16
30
2
Among Group Variation
x
x-x
3
5
7
15
x=5
-2
0
2
0
(x - x)
4
0
4
8
2
2
(x - x) n
20
0
20
40
SS Among = 40
Degrees of Freedom
 Among
group degrees of freedom equals
number of groups minus one (k-1)
 Within group degrees of freedom equals
the number of groups times the number
within each group minus one k(N-1)
 Total group degrees of freedom equals
the total number of subjects minus one
(kN-1) OR Degrees of freedom among
plus the degrees of freedom within
Analysis of Variance
Source
SS
df
MS F
Among
40
2
20
Within
30
12
2.5
Total
70
14
8
Analysis of Variance
Example exhibits significant
difference at the alpha = .01
level of significance
F.05 WITH 2 AND 12 df. = 3.88
F.01 WITH 2 AND 12 df. = 6.93
Result: 8 > 6.93; Reject Ho