at Looking data: relationships
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Correlation
IPS chapter 2.2
Copyright Brigitte Baldi 2005
Objectives (BPS chapter 2.2)
Correlation
The correlation coefficient r R does not distinguish x and y R has no units R ranges from -1 to +1 Influential points
The correlation coefficient "r"
The correlation coefficient is a measure of the direction and strength of a linear relationship between two quantitative variables x and y.
Student (i) IQ score (x) x1= 110 x2= 124 x3= 126 SelfConcept Score (y) y1= 58 y2= 72 y3= 80
i=1 i=2 i=3
x = 120, s x = 8.72
Correlation can only be used to describe quantitative variables. Categorical variables dont have means and standard deviations.
y = 70, s y = 11.14 What is the value of r?
r does not distinguish x & y
The correlation coefficient, r, treats x and y symmetrically.
r = -0.75
r=?
r does not distinguish x & y
The correlation coefficient, r, treats x and y symmetrically.
r = -0.75
r = -0.75
"Time to swim" is the explanatory variable here, and belongs on the x axis. However, in either plot r is the same (r=-0.75).
"r" has no unit
Changing the units of variables does not change the correlation coefficient "r", because we get rid of all our units when we standardize (get z-scores).
r = -0.75
z-score plot is the same for both plots
r = -0.75
"r" ranges from -1 to +1
"r" quantifies the strength and direction of a linear relationship between 2 quantitative variables.
Strength:howcloselythepoints followastraightline. Direction:ispositivewhen individualswithhigherXvalues tendtohavehighervaluesofY.
The Sign of r shows the direction of the Relationship
If r is positive, there is a positive x association and y. If r is negative, there is a negative association between x and y. Example 1: Suppose that x is height y is weight. There is a positive association between x and y because people of above-average height tend to be of above-average weight. Individual 1 (tall, heavy) ( Individual 2 (short, light)
x1 x y1 y )( ) = (+)(+) = + sx sy
( x2 x y 2 y )( ) = ()() = + sx sy
Example 2: Suppose that x is hours spending watching TV and y is GPA. There is a negative association between x and y because people who watch above-average hours of TV tend to have a below-average weight. Individual 1 (lot of TV, low GPA) ( Individual 2 (little TV, high GPA)
x1 x y1 y )( ) = (+ )() = sx sy
x2 x y 2 y )( ) = ()(+) = sx sy
(
When variability in one or both variables decreases, the correlation coefficient gets stronger ( closer to +1 or -1).
Correlation only describes linear relationships
No matter how strong the association, r does not describe curved relationships.
Influential points
Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers.
Just moving one point away from the general trend here decreases the correlation from -0.91 to -0.75
An example
If women always marry men 2 years older than themselves, what is the correlation of the ages between husband and wife? r=1
Thought quiz on correlation
1. Why is there no distinction between explanatory and response variable in correlation? 2. Why do both variables have to be quantitative? 3. How does changing the units of one variable affect a correlation? 4. What is the effect of outliers on correlations? 5. Why doesnt a tight fit to a horizontal line imply a strong correlation?