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
Turnout By % College
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.
Scattergrams
Turnout By % College
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
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.
Scattergrams
Turnout By % College
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
Vertical (Y) axis is voter turnout. Scores range from 44.1% to 70.4% and
increase from bottom to top
Scattergrams: Regression Line
Turnout By % College
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
A single straight line that comes as close as possible to all data points.
Indicates strength and direction of the relationship.
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
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
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
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
Scattergrams Inspection of the scattergram should
always be the first step in assessing the correlation between two I-R variables
Turnout By % College
43
48
53
58
63
68
73
15 17 19 21 23 25 27 29 31 33 35
% College
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 The scores on each
variable are displayed in table format: Y = Turnout X = Years of
Education
City X Y
A 11.9 55
B 12.1 60
C 12.7 65
D 12.8 68
E 13.0 70
Example of Computation Sums are
needed to compute b, a, and Pearson’s r.
X Y X2 Y2
XY
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
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.
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