Correlation
CJ 526 Statistical Analysis in
Criminal Justice
Correlation and Prediction
1. If a relationship exists between two
variables
2. Usually used with ex post facto designed
3. No manipulation of an IV by the
researcher
Requirements for Correlation
1. Requires two scores for each unit of
analysis:
1. X
2. Y
Represented by a scatterplot
Graphical representation of relationship between the
two variables
GPA
ACT
Characteristics of a Relationship
1. Direction (sign)
1. +: Positive
2. -: Negative
Direction
1. Positive
As one variable increases, the other increases
Scatterplot goes to the right
Negative
As one variable increases, the other decreases
Scatterplot goes to the left
Magnitude
1. Strength of a relationship
Closer to 1 or to -1, stronger the relationship
Less predictive error
Closer to 0, the weaker the relationship
More error in prediction
Magnitude -- continued
Zero correlation
1. Result of no systematic relationship between X
and Y
2. Knowing X would be of no value in predicting Y
Magnitude -- continued
Perfect correlations can be positive or
negative
Strong relationships can be either positive or
negative
The negative sign only indicates the direction
of the relationship, not the strength or
ability to predict
Interpretation Heuristic for
Magnitude: Positive Correlation
Correlation
Coefficient Range
0 to 0.4
0 to -.4
0.4 to 0.8
-.4 to -.8
0.8 to 1.0
-.8 to -1.0
Description
No to weak relationship
Moderate relationship
Strong relationship
Form
1. Form:
Linear and non-linear relationships
Linear: every change in X is accompanied
by a corresponding change in Y
Nonlinear Relationship
1. No linear relationship
1. A change in X does not correspond to any
predictable change in Y
Example: 0 correlation
Parabola
Nonlinear Relationships
1. Exponential
1. Time and retention
Retention
Time
Performance
Arousal
Use of Correlation
1. Reliability
Test-retest and split-half
2. Personality
Correlating test scores on personality tests:
scales with similar traits should have high
correlations, and scales with differing or
opposite traits should have lower
correlations
Pearson Product-Moment
Correlation
1. Measures the direction and strength of
the linear relationship between two
variables
Pearson Product-Moment
Correlation -- continued
degree to which X and Y vary together
(covariance)
1. divided by the variations in X and the variation in
Y
2. See p. 462 for the computational formula
Correlation and Causality
Correlation does not imply causality
Cause requires 3 criteria: (1) temporal; (2)
correlation; and (3) nonspuriousness—
relationship cannot be explained by a
third variable
Cause: relationship between x (presumed
cause) and Y (effect)
Poverty and Crime
1. Poverty and crime are related, as arrest
statistics indicate
Does poverty “cause” crime? There are poor
people who do not commit crime and
non-poor people who do
Factors Affecting Pearson
Correlation
Restricted range
1. Could overestimate or underestimate
Example
The correlation between ACT and GPA will be much
lower if you look at the range between 24 and 30
Interpreting Correlation in
Terms of Variance
Coefficient of Determination
1. Proportion of variance of Y that is explained or
accounted for by the variance of X
r squared
Coefficient of Nondetermination
Proportion of variance of Y that is not
explained or accounted for by the
variance of X
r
0.0
.2
.4
.6
.8
.9
r2
0.0
.04
.16
.36
.64
.81
%
Explained
0
4
16
36
64
81
1 - r2
1.0
.96
.84
.64
.36
.19
%
Unexplained
100
96
84
64
36
19
SPSS Procedure Graphs
• Use to generate scatterplot
– Determine whether the relationship is linear
• Graphs, Scatter
– Simple
• Define
SPSS Procedure Correlate
• Analyze, Correlate, Bivariate
– Move variables over
– Options
• Statistics
– Means and standard deviations
SPSS Procedure Correlate
Output
• Descriptive Statistics
–
–
–
–
Variables
Mean
Standard Deviation
N
• Correlations
– Pearson Correlation
– Sig (2-tailed)
– N
Hypothesis Tests With Pearson
Correlations
•
•
•
•
H0: The population correlation is zero
H1: The population correlation is non-zero
 (rho)
df = N - 2
Report Writing
• A correlation for the data revealed that
population and crime rate were
significantly related, (r = .97, p < .01).