1. Math 4030 – 12a Correlation

2. Correlation Two random variables, X and Y, both continuous numerical;
Correlation exists when the value of one variable go “consistently” up or down with the change of the other variable.
Correlation coefficient: r  [-1,1]

3. x y x2 y2 xy x1 y1 … xn yn
xi
yi
xi2
yi2
xiyi
Calculation: or

4. Meaning of r values: r = 0.5 r = 0.01 b = 0.8 b = 0.9 r = - 0.95

5. r vs. b: r and b have the same sign;
b is the slope of the linear relationship;
r is the strength of the linear relationship;
r  [-1,1], b  (-, +).

6. Testing about the normal population correlation coefficient :
Distribution of sample statistic r?
Fisher Z transformation:
r  (-1, 1)  Fisher-  (- , )
If joint distribution of (X,Y) is approximately bivariate normal, then

7. Test statistic for H0:  = 0

8. Confidence interval for :
Confidence interval for Fisher-Z score:
Solve the two boundary value for  using relationship

9. Strength vs. significance of the correlation:
the significance, given by P-value, depends on the statistical evidence. When small, the correlation (despite of the strength) exists.
the strength, given by the r value, is meaningful only it is supported by statistical significance.

10. Correlation does not mean causation!
Final Remark:
Correlation does not mean causation!