Chapter 15
Association Between Variables
Measured at the Interval-Ratio
Level
Chapter Outline
 Interpreting the Correlation
Coefficient: r 2
 The Correlation Matrix
 Testing Pearson’s r for Significance
 Interpreting Statistics: The Correlates
of Crime
Scattergrams
 Scattergrams have two dimensions:
 The X (independent) variable is arrayed
along the horizontal axis.
 The Y (dependent) variable is arrayed
along the vertical axis.
Scattergrams
 Each dot on a scattergram is a case.
 The dot is placed at the intersection
of the case’s scores on X and Y.
Scattergra ms
 Shows the relationship between %
College Educated (X) and Voter Turnout
(Y) on election day for the 50 states.
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams
Horizontal X axis - % of population of a
state with a college education.
Scores range from 15.3% to 34.6%
and increase from left to right.
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams
 Vertical (Y) axis is voter turnout.
 Scores range from 44.1% to 70.4% and
increase from bottom to top
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams: Regression Line
 A single straight line that comes as close as
possible to all data points.
 Indicates strength and direction of the
relationship.
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams:
Strength of Regression Line
 The greater the extent to which dots are clustered
around the regression line, the stronger the
relationship.
 This relationship is weak to moderate in strength.
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams:
Direction of Regression Line
 Positive: regression line rises left to right.
 Negative: regression line falls left to right.
 This a positive relationship: As % college
educated increases, turnout increases.
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
Scattergrams
 Inspection of the scattergram should
always be the first step in assessing the
correlation between two I-R variables
Turnout By % College
73
68
63
58
53
48
43
15
17
19
21
23
25
% College
27
29
31
33
35
The Regression Line: Formula
 This formula defines the regression line:
 Y = a + bX
 Where:
 Y = score on the dependent variable
 a = the Y intercept or the point where the
regression line crosses the Y axis.
 b = the slope of the regression line or the
amount of change produced in Y by a unit
change in X
 X = score on the independent variable
Regression Analysis
 Before using the formula for the regression line, a
and b must be calculated.
 Compute b first, using Formula 15.3 (we won’t do
any calculation for this chapter)
Regression Analysis
 The Y intercept (a) is computed from
Formula 15.4:
Regression Analysis
 For the relationship between % college
educated and turnout:
 b (slope) = .42
 a (Y intercept)= 50.03
 Regression formula: Y = 50.03 + .42 X
 A slope of .42 means that turnout increases
by .42 (less than half a percent) for every
unit increase of 1 in % college educated.
 The Y intercept means that the regression
line crosses the Y axis at Y = 50.03.
Predicting Y
 What turnout would be expected in a state
where only 10% of the population was
college educated?
 What turnout would be expected in a state
where 70% of the population was college
educated?
 This is a positive relationship so the value
for Y increases as X increases:
 For X =10, Y = 50.3 +.42(10) = 54.5
 For X =70, Y = 50.3 + .42(70) = 79.7
Pearson correlation coefficient
 But of course, this is just an estimate of
turnout based on % college educated, and
many other factors also affect voter
turnout.
 How much of the variation in voter turnout
depends on % college educated? The
relevant statististic is the coefficient of
determination (r squared), but first we
need to learn about Pearson’s correlation
coefficient (r).
Pearson’s r
 Pearson’s r is a measure of association for I-R
variables.
 It varies from -1.0 to +1.0
 Relationship may be positive (as X increases, Y
increases) or negative (as X increases, Y decreases)
 For the relationship between % college educated and
turnout, r =.32.
 The relationship is positive: as level of education
increases, turnout increases.
 How strong is the relationship? For that we use R
squared, but first, let’s look at the calculation process
Example of Computation
 The computation and interpretation of a, b,
and Pearson’s r will be illustrated using
Problem 15.1.
 The variables are:
 Voter turnout (Y)
 Average years of school (X)
 The sample is 5 cities.
 This is only to simplify computations, 5 is much
too small a sample for serious research.
Example of Computation
City
X
Y
A
11.9
55
B
12.1
60
C
12.7
65
D
12.8
68
E
13.0
70
 The scores on each
variable are
displayed in table
format:
 Y = Turnout
 X = Years of
Education
Example of Computation
Y2
XY
X
Y
X2
11.9
55
141.61
3025
654.5
12.1
60
146.41
3600
726
12.7
65
161.29
4225
825.5
12.8
68
163.84
4624
870.4
13.0
70
169
4900
910
62.5
318
782.15
20374
3986.4

Sums are
needed to
compute b, a,
and Pearson’s
r.
Interpreting Pearson’s r
 An r of 0.98 indicates an extremely strong
relationship between average years of
education and voter turnout for these five
cities.
 The coefficient of determination is r2 = .96.
Knowing education level improves our
prediction of voter turnout by 96%. This is
a PRE measure (like lambda and gamma)
 We could also say that education explains
96% of the variation in voter turnout.
Interpreting Pearson’s r
 Our first example provides a more
realistic value for r.
 The r between turnout and % college
educated for the 50 states was:
 r = .32
 This is a weak to moderate, positive
relationship.
 The value of r2 is .10.
Percent college educated explains
10% of the variation in turnout.