Overview of Correlation
 What
is a Correlation?
 Correlation Coefficients
 Coefficient of Determination
 Test for Significance
 Correlation and Causality
 Partial and Part Correlations
What is a Correlation?
Degree of linear relationship between
 Each individual is measured on both
What is a Correlation?
 Comparison
of the way scores deviate
from their means on the two variables
 Standardized covariance
Cross-Product Deviation
Find the difference between each
person’s scores and the mean of the
variable (deviation).
 For each person, multiply the two
deviations together.
 Do the deviations tend to go in the
same direction?
Add up all the cross-product
deviations and average them.
 The more covariance, the more the two
variables go together, or co-vary.
 Covariance is not standardized, so it’s
hard to interpret.
Pearson r
 Standardized
 Used for two interval/ratio variables
 Varies from -1 to +1
Pearson r
Absolute value indicates strength of
 .1
- small
 .3 - medium
 .5 - large
Pearson r
Sign indicates direction of correlation
 Positive:
increases on one variable
correspond to increases on the other
 Negative: increases on one variable
correspond to decreases on the other
Other Correlation Coefficients
 Ordinal
Spearman rho or Kendall’s tau
 Dichotomous
variable with
interval/ratio variable
 Point
biserial r (discrete dichotomy)
 Biserial r (continuous dichotomy)
Other Correlation Coefficients
 Two
dichotomous variables
 Phi
About Dichotomous Variables
Dichotomous variables are usually at
the nominal level.
 Numbers are assigned to the two
categories in an arbitrary way.
 The way the numbers are assigned
determines the sign of the correlation
Review Question!
How is covariance related to the
correlation coefficient?
Coefficient of Determination
Measures proportion of explained
variance in Y based on X
 r2
Testing r for Significance
Null hypothesis is usually that r is zero
in the population.
 One tailed vs. two-tailed
 Appropriate
types of data
 Independent observations
 Normal distributions
 Linear relationship
Example APA format
The Pearson r was computed between
rated enjoyment of frog legs and level
of neuroticism. The correlation was
statistically significant, r(58) = .28, p =
Review Question
If r = .28, then r2 = .08. What does the .08
Review Question!
If p = .03, what probability does the .03
represent? There is a 3% chance of…..?
Correlation and Causality
A correlation by itself does not show
that one variable causes the other.
 A correlation may be consistent with a
causal relationship.
The Third Variable Problem
A correlation between X and Y could
be caused by a third variable
influencing both X and Y.
The Directionality Problem
A correlation between X and Y could
be a result of X causing Y or Y causing
Partial Correlation
 Used
to “partial out” the effects of a third
variable (X2) on the relationship between X1
and Y
 Correlation between X1 and Y with the
influence of X2 removed from both X1 and Y
Partial r2
Interpreting Partial
 Compare
the simple bivariate
correlation to the partial correlation.
 If the partial correlation is lower, it
suggests that X2 is mediating the
relationship between X1 and Y.
Part Correlation
 Also
called: semi-partial correlation
 Correlation between X1 and Y with the
influence of X2 (and other predictor
variables) removed from just X1
 Indicates amount of unique variance in
Y explained by X1
 Used in Multiple Regression Analysis
Part r2
Partial r2
Choosing Stats
Patrons at a bar are randomly assigned to one
of three information conditions. In one
condition, they taste a beer without being given
any information about it. In a second condition,
they are told that it is an inexpensive brand of
beer. In a third condition, they are told that it is
an expensive brand of beer. Their ratings of the
taste quality are compared across the three

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