Correlation
• Correlation: the mathematical extent to which two
variables are related to each other
– Correlation refers to both a type of research design and
a descriptive statistical procedure.
– Generally performed between two scores obtained from
the same source
Correlation Coefficient
• Correlation Coefficient:
number between +1 and -1 that
represents the strength and
direction of the relationship
between two variables
• Correlations that are closer to
+1 and –1 are stronger and are
better able to accurately predict
Types of Correlation Coefficients
• Pearson r: both variables
are measured at an
interval/ratio level
• Spearman rho: used when
the measurement of at least
one variable is ordinal
(scores on the other variable
must be converted to ranks)
Positive Correlations
• Positive Correlation: a
correlation that is a greater
than zero, but less than +1
• Indicates that high scores on
one variable are associated
with high scores on another
variable
• The values of the variables
increase and decrease
together.
Negative Correlations
• Negative Correlation: a
correlation coefficient whose
value is between 0 and -1
• Indicates that there is an
inverse relationship between
the two sets of scores
• A high score on X is related
to a low score on Y, and vice
versa
Linear Relationships
Freshman GPA
• Linear Relationship: a
condition wherein the
relationship between
two variables can be
best described by a
straight line (the
regression line or the
line of best fit)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
300
400
500
600
SAT Score
700
800
Scatterplots
• Scatterplot: provides a visual representation of the
relationship between variables
• Each point represents paired measurements on two
variables for a specific individual
Understanding the Pearson Product
Moment Correlation Coefficient
• Pearson r: represents the
extent to which individuals
occupy the same relative
position in two distributions
• Definitional Equation:
Σz x z y
r =
N
• Important Reminder:
– Σz2 = N
Interpreting the Correlation Coefficient
• Coefficient of Determination (r2): the proportion of
variance in one variable that can be described or explained
by the other variable
• Coefficient of Nondetermination (1 - r2): the proportion of
variance in one variable that cannot be described or
explained by the other variable
Correlation Matrices
• Tables of correlations are generated when more than two
variables are involved.
• A Correlation Matrix is a table in which each variable is
listed both at the top and at the left side, and the correlation
of all possible pairs of variables is shown inside the table
• An asterisk identifies significant correlations.
Correlations
Fres hman GPA
Hours Worked per Week
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Hours
Fres hman
Worked
GPA
per Week
1
-.693**
.
.004
15
15
-.693**
1
.004
.
15
15
**. Correlation is s ignificant at the 0.01 level (2-tailed).
Caution: Spurious Correlations
• Spurious Correlations: a correlation coefficient that
is artificially high or low because of the nature of the
data or method for collecting the data
• Common Causes of Spurious Correlations:
–
–
–
–
–
–
A nonlinear relationship
Truncated range
Sample Size
Outliers
Multiple Populations
Extreme Scores
Caution: No Causality
•
•
Correlations only tell us that two
variables are related; they do not
determine causality
Four Possible Explanations:
1. X
Y (Temporal Directionality)
2. Y
X (Temporal Directionality)
3. X
Y (Bidirectional Causation)
4. Z
X and Y (Third Variable
Problem)
Computing the Correlation Coefficient
Using SPSS
• Analyze  Correlate  Bivariate
• Select variables to be correlated in the left side of
the Bivariate Correlations window and move them
to the right side
• Select the appropriate correlation coefficient
• Check two tailed and flag significant correlations 
click OK
Interpreting the Output
Correlations
Freshman GPA
SAT Score
Hours Studied per Week
Hours Worked per Week
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Hours
Hours
Freshman
Studied
Worked
GPA
SAT Score per Week per Week
1
.685**
.548*
-.693**
.
.005
.034
.004
15
15
15
15
.685**
1
.041
-.612*
.005
.
.884
.015
15
15
15
15
.548*
.041
1
-.398
.034
.884
.
.142
15
15
15
15
-.693**
-.612*
-.398
1
.004
.015
.142
.
15
15
15
15
**. Correlation is s ignificant at the 0.01 level (2-tailed).
*. Correlation is s ignificant at the 0.05 level (2-tailed).
Creating a Scatterplot
•
•
•
•
•
•
•
•
Graphs  Scatter
Click Simple  Click Define
Move the criterion variable to the Y axis box
Move the predictor variable to the X axis box
Click OK
Double-click on the chart to edit it.
Click Fit Line at Total.
Click OK
Reading Scatterplots
4.0
4.0
3.5
3.5
3.0
Freshman GPA
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
-10
1.0
0
10
20
Hours Worked per Week
30
40
0
10
20
Hours Studied per Week
30
40
Linear Regression
• An important use of the
correlation coefficient is the
ability to predict one set of
scores from another.
• If we know the score on one
variable, we can use that
score to predict someone’s
score on the correlated
variable.
The Regression Line
4.0
Freshman GPA
• Line of Best Fit:
minimizes the
distance between
each individual
point and the
regression line
3.5
3.0
2.5
2.0
1.5
1.0
300
400
500
600
SAT Score
700
800
The Regression Equation
• Equation: Y’ = aY + bY(X)
• Where
Y’ = the predicted score of Y
based on a known value
of X
aY = the intercept of the
regression line
bY = the slope of the line
X = the score being used as
the predictor
In English Please…
• Slope: how much variable Y
changes as the values of
variable X change one unit
• Intercept: the value of variable
Y when X = 0
• Predictor Variable: the variable
X which is used to predict the
score on variable Y (antecedent
or independent variable)
• Criterion Variable: the variable
that is predicted (dependent
variable)
Linear Regression Using SPSS
• Analyze  Regression  Linear
• Click on the criterion variable and move it to the Dependent
box
• Click on the predictor variable and move ot to the
Independent(s) box
• Click Statistics  check Descriptives  make sure that
Estimates and Model fit are also selected
• Click Continue
• Click OK
Interpreting the Output
ANOVAb
Model
1
Regress ion
Res idual
Total
Sum of
Squares
2.862
6.674
9.536
df
1
13
14
Mean Square
2.862
.513
F
5.575
Sig.
.034 a
a. Predictors : (Constant), Hours Studied per Week
• The F value
in the ANOVA box indicates whether the predictor variable was a
b. Dependent Variable: Fres hman GPA
significant predictor of the criterion variable.
Coefficientsa
Model
1
(Cons tant)
Hours Studied per Week
Uns tandardized
Coefficients
B
Std. Error
1.735
.395
.060
.025
Standardized
Coefficients
Beta
.548
t
4.388
2.361
Sig.
.001
.034
a. Dependent Variable: Fres hman GPA
• The unstandardized coefficient for the constant reflects the Y intercept of the
regression equation.
•The unstandardized coefficient for the predictor variable reflects the slope of the line.
•The regression equation for this example would be Y’ = 1.735 + .06X