1. Association Between Variables Measured at the Interval-Ratio Level
Chapter 15
Association Between Variables Measured at the Interval-Ratio Level

2. Chapter Outline Interpreting the Correlation Coefficient: r 2
The Correlation Matrix
Testing Pearson’s r for Significance
Interpreting Statistics: The Correlates of Crime

3. 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.

4. 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.

5. Scattergra ms
Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.

6. 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.

7. Scattergrams
Vertical (Y) axis is voter turnout.
Scores range from 44.1% to 70.4% and increase from bottom to top

8. Scattergrams: Regression Line
A single straight line that comes as close as possible to all data points.
Indicates strength and direction of the relationship.

9. 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.

10. 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.

11. Scattergrams
Inspection of the scattergram should always be the first step in assessing the correlation between two I-R variables

12. 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

13. 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)

14. Regression Analysis
The Y intercept (a) is computed from Formula 15.4:

15. Regression Analysis
For the relationship between % college educated and turnout:
b (slope) = .42
a (Y intercept)= 50.03
Regression formula: Y = 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 =

16. 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 = (10) = 54.5
For X =70, Y = (70) = 79.7

17. 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).

18. 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

19. 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.

20. 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

21. 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

22. 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.

23. 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.