1. Measures of Association: Pairwise Correlation

2. Covariance
Covariance is a measure of association between two random variables. OR
Cov(x,y) = (average products of XY) –
(average of X)(average of Y)

3. Calculating CovarianceX Y XY 1 9 2 15 30 4 8 32 5 10 3 11 33
Mean= 2.4
Mean= 9.6
Mean=22.8
Cov= 22.8 – (2.4)(9.6) = -0.24

4. Correlation Symbol and Formula
ρ for the population
r for the sample
ρ = σxy /(σx σy)

5. Calculating CorrelationsX Y XY 1 9 2 15 30 4 8 32 5 10 3 11 33
Mean= 2.4
Mean= 9.6
Mean=22.8
St. deviation = 1.72
St. deviation = 3.32
Cov= 22.8 – (2.4)(9.6) = -0.24
r= / [(1.02)(3.32)] =

6. Interpreting Results The rho Statistic is:
Independent of what scales X and Y are measured on, and
Bounded by -1 and 1, where
0 means no relationship
-1 means a perfect negative relationship (as X increases Y decreases)
1 means a perfect positive relationship (as X increases Y increases)
Rho has its own distribution and significance

7. Rho = 1

8. Rho = -1

9. Rho = .96

10. Limitations
Only gives you information about linearity of the relationship. Otherwise put, a strong correlation is indicative of a purely mathematical relationship, not a causal one.
However, looking for high correlations among variables is a very good way to start testing your ideas abut whether variables have causal effects on each other or not.

11. Rho = 0

12. Rho= .96

13. Same Data with Narrower Range

14. Magnitude of Relationships