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USES OF CORRELATION
• Test reliability • Test validity• Predict future scores (regression)• Test hypotheses about relationships between
variables
Doing Correlational Research
• Any of the descriptive methods can be used (observation, survey, archival, physical traces).
• Must have pairs of scores
Doing Correlational Research
• The pairs of scores should be independent.• Both variables should be normally
distributed.• If there is a relationship between variables, it
should be linear.
Correlation and Cause
• A correlation by itself does not show that one variable causes the other.
• A correlation is 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.
• example: The use birth control is correlated with the number of electrical appliances in the household.
The Directionality Problem
• A correlation between X and Y could be a result of X causing Y or Y causing X
• example: Amount of TV watching and the level of aggression are correlated.
The Cross-Lagged Panel
• Design used to determine direction of cause• Measure both variables at two different
points in time• Cause cannot work backwards in time
Time 1 Time 2variable X variable X
variable Y variable Y
The Correlation Coefficient
• Strength of relationship– 0 means no relationship at all– -1 or +1 means perfectly related
• Direction of relationship– positive: variable X increases as variable Y
increases– negative: variable X decreases as variable Y
increases
The Scatterplot
X
Y
low high
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positive r
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onegative r
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low high
high
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zero r
X
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low high
high
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Non-linearrelationship
Correlation Coefficients
X data Y data Coefficientinterval/ratio interval/ratio Pearson rordinal ordinal Spearman rhodichotomous interval/ratio Point Biserialdichotomous dichotomous Phi
dichotomous: having only two values
More on Dichotomous Variables
• With dichotomous variables, whether r is negative or positive depends on how the numbers were assigned
More on Dichotomous Variables
• If the correlation between gender and GPA is positive, it could mean that– females have higher GPAs, if males were 1’s and
females were 2’s– males have higher GPAs, if females were 1’s and
males were 2’s
Pearson r formula
r = zxzy
Nzx = z - score on x
zy = z - score on y
N = # of individuals
Computation of Pearson r
Example: Compute the correlationbetween scores on Exam1 and Exam2.Student Exam1 Exam21 97 862 82 953 74 794 89 955 93 90
STEP 1:Convert the x scores to z-scores.
Exam1 x- (x-)2 zx
97 10 100 +1.2282 -5 25 -.6174 -13 169 -1.5989 2 4 +.2493 6 36 +.73
=87 =334x = 334/5 = 8.17
STEP 2:Convert the y scores to z-scores.
Exam2 y- (y-)2 zy
86 -3 9 -.5095 6 36 +1.0079 -10 100 -1.6695 6 36 +1.0090 1 1 +.17
=89 =182y = 182/5 = 6.03
STEP 3: Multiply the z-scores.
zx zy zxzy
+1.22 -.50 -.61-.61 +1.00 -.61-1.59 -1.66 +2.64+.24 +1.00 +.24+.73 +.17 +.12
STEP 4: Add up the zxzy products.
zxzy
-.61-.61+2.64+.24+.12
= 1.78
STEP 5: Divide zxzy by N.
r =1.78
5.36
Coefficient of Determination
• Measures proportion of explained variance in Y based on X.
• Square r to get r2. Example: r = .36 r2 = .13
We can explain 13% of the differences in Exam 2 scores by knowing Exam 1
scores.
What Could a Low r Mean?
• Lack of a relationship.• Unreliable measurement.• Non-linear relationship.• Restricted range : full range of scores not
measured on both variables
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