Analysis of Variance
(ANOVA)
(Click icon for audio)
Dr. Michael R. Hyman, NMSU
2
Analysis of Variance
Sum of Squares
SStotal  SSwithin  SSbetween
3
Analysis of Variance
Sum of Squares Between
n
SS between   n j ( X j  X )
2
j 1
4
Analysis of Variance
Sum of Squares Between
X j= individual scores, i.e., the ith observation or
test unit in the jth group
X = grand mean
nj = number of all observations or test units in a
group
5
Analysis of Variance
Sum of Squares Within
n
c
SS within   ( X ij  X j )
i  1 j 1
6
2
Analysis of Variance
Sum of Squares Within
X
piij= individual scores, i.e., the ith observation or
test unit in the jth group
pi = grand mean
X
n = number of all observations or test units in a
group
c = number of jth groups (or columns)
7
Analysis of Variance
Sum of Squares Total
n
c
SStotal   ( X ij  X )
i  1 j 1
8
2
Analysis of Variance
Sum of Squares
X
piij = individual scores, i.e., the ith observation or
test unit in the jth group
pi = grand mean
X
n = number of all observations or test units in a
group
c = number of jth groups (or columns)
9
Analysis of Variance
Mean Squares Between
MS between
SS between

c 1
10
Analysis of Variance
Mean Square Within
MS within
SS within

cn  c
11
Analysis of Variance
F-Ratio
Variance  between  groups
F
Variance  within  groups
12
Analysis of Variance
F-Ratio
MSbetween
F
MS within
13
ANOVA Summary Table
Source of Variation
• Between groups
• Sum of squares
– SS between
• Degrees of freedom
– c-1 where c=number of groups
• Mean squared-MS between
– SS between / c-1
14
ANOVA Summary Table
Source of Variation
• Within groups
• Sum of squares
– SS within
• Degrees of freedom
– cn-c where c=number of groups, and
n = number of observations in a group
• Mean squared – MS within
– SS within / cn-c
15
ANOVA Summary Table
Source of Variation
• Total
• Sum of Squares
– SStotal
• Degrees of Freedom
– cn-1 where c = number of groups, and
n = number of observations in a group
MS BETWEEN
F
MS WITHIN
16
Examples
17
Example #1
18
19
Example #2
20
Test Market Pricing Experiment
Sales in Units (thousands)
Test Market A, B, or C
Test Market D, E, or F
Test Market G, H, or I
Test Market J, K, or L
Mean
Grand Mean
Regular Price
$.99
Reduced Price
$.89
Cents-Off Coupon
Regular Price
130
118
87
84
145
143
120
131
153
129
96
99
X1=104.75
X=119.58
X2=134.75
X1=119.25
Example #3
21