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University of North Texas Dr. J. Kyle Roberts © 2004
Unit 2: Bivariate Relationships
Lesson 2: Covariance and Pearson r
EDER 6010: Statistics for Educational Research
Dr. J. Kyle Roberts
University of North Texas
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University of North Texas Dr. J. Kyle Roberts © 2004
Are The VariablesCorrelated???
Covariance
1
))((
n
YYXXCOV ii
XY
“The average cross-product of the deviation scores.”
Answers 2 Questions:
1. Is there any relationship between X and Y?
2. If there is a relationship, is it positive or negative?
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University of North Texas Dr. J. Kyle Roberts © 2004
“Rules” for Covariance
2. Covariance will be 0 (zero) when the sum of the cross-products is 0 (e.g., IQ and shoe size)
3. The “sign” of the covariance tells the “direction” of therelationship Do they “Covary”
Together???
1. There are no “bounds” for values for the covariance
Are The VariablesCorrelated???
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University of North Texas Dr. J. Kyle Roberts © 2004
Pearson r
yx
xyXY SDSD
COVr
*
“The average cross-product of the standardized deviation scores.”
Answers 3 Questions:
1. Is there any relationship between X and Y?
2. If there is a relationship, is it positive or negative?
3. How well does one line describe the data?
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University of North Texas Dr. J. Kyle Roberts © 2004
Pearson r“How well does a single line represent my data?”
r = .85 r = -.25 r = 1.0
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University of North Texas Dr. J. Kyle Roberts © 2004
Pearson r
X0011
Y0101
r = .00
3
25.)25.()25.(25.3
)5.1)(5.1()5.0)(5.1()5.1)(5.0()5.0)(5.0(
XY
XY
COV
COV
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University of North Texas Dr. J. Kyle Roberts © 2004
“Guessing” Pearson r
Height and weight
Moderate PositiveSES and Math Achievement
High NegativeTobacco use and Life Expectancy
High PositiveGPA and SAT
Near ZeroShoe size and IQ
Moderate to High Positive
Parameters Pearson r
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University of North Texas Dr. J. Kyle Roberts © 2004
“Rules” for Pearson r
2. A zero value for the Pearson r means that there is NOrelationship between the two variables of interest
3. Pearson r’s cannot be compared between two studies
1. The “bounds” for Pearson r are between +1.0 and -1.0
4. The Covariance determines the sign of the Pearson r
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University of North Texas Dr. J. Kyle Roberts © 2004
Square Before You Compare
R2 = .25 R2 = .36
Study 1
Pearson r betweenGPA and GREr = .50
Study 2
Pearson r betweenIQ and GREr = .60
“Study 2 explains 11% more variance than Study 1”
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University of North Texas Dr. J. Kyle Roberts © 2004
A Little Practice
r = 1.0 r = .832
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University of North Texas Dr. J. Kyle Roberts © 2004
Unit 2: Bivariate Relationships
Lesson 2: Covariance and Pearson r
EDER 6010: Statistics for Educational Research
Dr. J. Kyle Roberts
University of North Texas
