1. Correlation
A measure of the strength of the linear association between two numerical variables.

2. Sample Covariance
Measure of the co-variability between two numerical variables.

3. Sample Correlation
The sample covariance scaled to account for variation in the x’s and y’s.

4. Properties
The value of the correlation coefficient, r, is always between
–1: a perfect negative linear relationship
+1: a perfect positive linear relationship

5. Properties
r = 0: There is no linear relationship between the two numerical variables.
Random scatter
There could be a relationship, but not one that is linear.

6. Properties The correlation coefficient, r, does not have any units.
Changing the scales of the numerical variables will not change the value of the correlation coefficient.

7. CO2 and Temperature

8. CO2 and Temperature
There is a strong, positive linear association between the carbon dioxide concentration and the temperature.

9. CO2 and Temperature
Is the linear association between the carbon dioxide concentration and temperature statistically significant?

10. Step 1: Hypotheses

11. Step 2: Test statistic

12. Step 3: Decision
Reject the null hypothesis because the P-value is so small.

13. Step 4: Conclusion
There is a statistically significant linear association between carbon dioxide concentration and temperature.

14. Connection
The test for the statistical significance of correlation is exactly the same as the test for the statistical significance of the estimated slope.

15. Connection

16. Connection

17. Difference R2 can be interpreted as a % of total variation.
r has a sign (+/–) that matches the direction of the association and cannot be interpreted as a %.

18. JMP Analyze – Multivariate Methods – Multivariate
Y, Columns: CO2, Temp
Multivariate
Pairwise correlations

20. JMP Output JMP does not give you the value of the test statistic.
JMP does give a 95% confidence interval for the population correlation coefficient.

21. 95% Confidence Interval r = 0.8977 95% confidence interval on ρ.to
Note that this interval is not symmetric around the value of r.