1. Introduction to bivariate statistics: Covariance and Correlation
Basic concept of association of two interval/ratio variables

2. Basics of concept of “association”
Income and education are associated
Religiosity and political conservatism are associated
Illicit drug use and GPA are associated
Authoritarian parenting and conservative gender role attitudes are associated

3. One question, three answers
Is drug use and GPA associated?
Case 1:
GPA is measured as actual GPA.
Drug use is measured by categorizing those who ever used and those who never used.
Case 2:
GPA is measured as actual GPA.
Drug use is measured by asking the number of times used drugs in the last month.
Case 3:
GPA is measured in 4 intervals: 0-.99; ; , 3+.
Drug use is measured by categorizing those who ever used and those who never used.

4. How do we measure association?
If two interval-ratio level variables
Correlation (or covariance)
If an interval level and nominal variable
T-test - example
Variants of correlation
If two nominal-ordinal variables
Chi-square test - example

5. Covariance

6. Understanding the covariance
Covariance should represent the degree of association
To the extent that differences from the mean are in the same direction, covariance should be larger
The stronger the association is, the larger the covariance should be

7. Application ID Education Income Education: Comparison to the mean
Income: Comparison to the mean 1 8
15000
-4.4
-14300 2 12
23000
-0.4
-6300 3 11
16000
-1.4
-13300 4 18
65000
5.6
35700 5 16
39000
3.6
9700 6
11000
-18300 7 14
28000
1.6
-1300
38000
8700 9
32000
2700 10
26000
-3.4
-3300
Mean
12.4
29300

8. Covariance Two variables, x and y. When one is above the mean,
is the other one also above the mean?
x & y are the variables that we are interested in
xi & yi are the values of the variables for one individual
n is the sample size

9. Example

10. REFERENCE TO G&W
P strength and direction of the association.

11. How about IQ and Income?

12. Inspect the association

13. Which association is stronger?

14. Compute covariance (again)

15. Comparison Covariance between education and income is: 46,088.89
Covariance between IQ and income is: 72,977.78

16. Comparison
Unit of measurement of covariance between education and income is:
Years of education dollars
Unit of measurement of covariance between IQ and income is:
IQ points dollars
QUESTION: When we need something in universal units, how do we measure it? (think t- test, think effect size)

17. How do we compare
Try to express both quantities in terms of a single unit of analysis
CORRELATION
Divide covariance by the standard deviation of each variable

18. Education and Income

19. IQ and Income

20. Comparing correlations
Correlation between education and income is: .83
Correlation between IQ and income is: .54

21. Correlation - formula Two variables, x and y.
When one is above the mean,
is the other one also above the mean?
x & y are the variables that we are interested in
xi & yi are the values of the variables for one individual
n is the sample size

22. Hypothesis testing with a correlation
The test is similar to a t-test. (G&W )
Null hypothesis: r=0
Alternative hypothesis: One tail or two tail

23. IMPORTANT INTERPRETATION of r
G&W p. 470 – 15.2

24. Conditions that make r small!
G&W p – read and ask questions if needed.