Measures of Association: Pairwise Correlation

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)

Calculating Covariance X 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

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

Calculating Correlations X 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= -0.24 / [(1.02)(3.32)] = -0.0709

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

Rho = 1

Rho = -1

Rho = .96

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.

Rho = 0

Rho= .96

Same Data with Narrower Range

Magnitude of Relationships