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Sociology
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SOCY 211
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Carl Keane
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ANALYSIS OF VARIANCE
- Often referred to as a “Comparison of Means” test
- Can be used when: Independent variable is measured at the nominal or ordinal
level, and the Dependent Variable is measured at the interval/ratio level.
- ANOVA
- A lot of our independent variables vary- such as happiness
- “Total Variation” = two components:
(1) the distance, or deviation, of raw scores from their group mean - this is known as
“Variation Within Groups,” and,
(2) the distance, or deviation, of group means from one another - this is known as
“Variation Between Groups.”
- Education and income example
- All have some overlap and within each group there will be some variations
- When you compare between groups you always use the mean
Sum of Squares
S =√ ∑(X – X)²
N
- We use all the values so that it is easier to compare
- When we have groups we have different sums of squares but each type represents
the sum of sum deviations from the mean
Total Sum of Squares
Within-Groups Sum of Squares
Between Groups Sum of Squares
Happiness Measured on a Scale from 1 to 10 (example)
Group A Group B Group C
8 x x
7 x x
5 x x
4 x x
24 X X
N(A)=4 N(B)=4 N(C)=4
Total N=12
Mean(A) = 24/4 = 6 Total Mean (Hypothetically) = 5
- WITHIN-GROUP VARIATION/DEVIATION (Raw scores for each group
member minus their group mean)
e.g., Group A 8 – 6 = +2
1 7 – 6 = +1
5 – 6 = -1
4 – 6 = -2
- Then do the same for individuals in Groups B and C – subtracting their individual
scores from their group means.
- BETWEEN-GROUP VARIATION/DEVIATION (Means for each group minus
the total mean)
e.g., Group A: 6 – 5 = +1
- Then do the same for Groups B and C
- TOTAL VARIATION/DEVIATION (Raw scores for each group member minus
the total mean)
e.g., Group A 8 – 5 = +3
7 – 5 = +2
5 – 5 = 0
4 – 5 = -1
- Then do the same for every member of Groups B and C.
Mean Squares
Total Sum of Squares - measures variation of individual scores about the total
mean
Within-Groups Sum of Squares - measures variation of individual scores about
their group mean
Between-Groups Sum of Squares - measures variation of group means about the
total mean
F RATIO
Between-Groups Mean Squares
F = --------------------------------------
Within-Groups Mean Squares
- Shows if our data is statistically significant
- The stronger the relationship between the two variables, the larger the value of f
(independent and dependent)
- If there is a big difference between the variance then there is a relationship
- If there is a weak association, the between group variance will be small
SUMMARY
Ratio of: Relationship
B-G Variance/Mean Squares
W-G Variance/Mean Squares
Large B-G Mean Squares
-------------------------- = Large F Ratio Strong
Small W-G Mean Squares
2 Small B-G Mean Squares
------------------------- = Small F Ratio Weak
Large W-G Mean Squares
Medium B-G Mean Squares
--------------------------- Moderate F Ratio Moderate
Medium W-G Mean Squares
Total Sum Within-Groups Between Groups
= +
of Squares Sum of Squares Sum of Squares
So,
Between-Groups Total Sum Within-Groups
= -
Sum of Squares of Squares Sum of Squares
2
- PRE measures of association with ANOVA is called ETA square (E )
- Makes use of the total sum of squares and group totals
- Total sum of squares is total variation in dependent variable
- Total sum of square is a measure of errors

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