correlational research studies describe a linear relationship between variables do not imply a...
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CORRELATIONAL RESEARCH STUDIES Describe a linear relationship between
variables Do not imply a cause-and-effect relationship Do imply that variables share something in
common
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CORRELATION COEFFICIENT Expresses degree of linear relatedness
between two variables Varies between –1.00 and +1.00 Strength of relationship is
Indicated by absolute value of coefficient Stronger as shared variance increases
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TWO TYPES OF CORRELATIONIf X… And Y…
The correlation is
Example
Increases in value
Increases in value
Positive or directThe taller one gets (X), the more one weighs (Y).
Decreases in value
Decreases in value
Positive or direct
The fewer mistakes one makes (X), the fewer hours of remedial work (Y) one participates in.
Increases in value
Decreases in value
Negative or inverse
The better one behaves (X), the fewer in-class suspensions (Y) one has.
Decreases in value
Increases in value
Negative or inverse
The less time one spends studying (X), the more errors one makes on the test (Y).
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WHAT CORRELATION COEFFICIENTS LOOK LIKE Pearson product moment correlation
rxy Correlation between variables x and y
Scattergram representation1. Set up x and y axes
2. Represent one variable on x axis and one on y axis
3. Plot each pair of x and y coordinates
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• When points are closer to a straight line, the correlation becomes stronger• As slope of line approaches 45°, correlation becomes stronger
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COMPUTING THE PEARSON CORRELATION COEFFICIENT
Where
rxy = the correlation coefficient between X and Y
= the summation sign
n = the size of the sample
X = the individual’s score on the X variable
Y = the individual’s score on the Y variable
XY = the product of each X score times its corresponding Y score
X2 = the individual X score, squared
Y2 = the individual Y score, squared
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AN EXAMPLE OF MORE THAN TWO VARIABLES
Grade Reading Math
Grade 1.00 .321 .039
Reading .321 1.00 .605
Math .039 .605 1.00
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INTERPRETING THE PEARSON CORRELATION COEFFICIENT “Eyeball” method
Correlations between
Are said to be
.8 and 1.0 Very strong
.6 and .8 Strong
.4 and .6 Moderate
.2 and .4 Weak
0 and .2 Very weak
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INTERPRETING THE PEARSON CORRELATION COEFFICIENT Coefficient of determination
Squared value of correlation coefficient Proportion of variance in one variable explained
by variance in the other Coefficient of alienation
1 – coefficient of determination Proportion of variance in one variable unexplained
by variance in the other
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RELATIONSHIP BETWEEN CORRELATION COEFFICIENT AND COEFFICIENT OF DETERMINATION
The increase in the proportion of variance explained is not linear
If rxy IsAnd
rxy2 Is
Then the Change
FromIs
0.1 0.01
0.2 0.04 .1 to .2 3%
0.3 0.09 .2 to .3 5%
0.4 0.16 .3 to .4 7%
0.5 0.25 .4 to .5 9%
0.6 0.36 .5 to .6 11%
0.7 0.49 .6 to .7 13%
0.8 0.64 .7 to .8 15%
0.9 0.81 .8 to .9 17%
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Doing it by hand
Make a Scatter Diagram
Student Hours Studied Test Score
A 0 52
B 10 92
C 6 75
D 8 71
E 6 64
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Change all scores to Z scores Figure the cross-product Add up the cross-produces Divide by the number of people in the study
r=∑ZxZy/N
Z=X-M/SD
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Prediction
Zy=(β)(Zx) So, if I know your score on x, I can predict
your score on Y.
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© 2009 Pearson Prentice Hall, Salkind.
A great link
http://www.hawaii.edu/powerkills/UC.HTM#C2