Correlation

In the above example the correlation of C1 and C2 = 0.983. The scatterplot shows a very definite linear relationaship, this is measured by a numerical summary: The correlation coefficient, it is symmetric in the two variables, is between -1 and +1, the larger it's absolute value the stronger the linear relation between the variables.

The two variables are usually measured in completely different units, the way to make the two comparable is to use our z-scale, ie deviations from the mean measured in standard deviations.

This gives the correlation coefficient defined as:

$$r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{ \sqrt{\sum (x_i-\bar{x})^2}\sqrt{\sum (y_i-\bar{y})^2 }}$$

$r$ is not affected by:

• interchanging the two variables
• adding the same number to all values of one variable
• multiplying all the values of one variable by the same positive number

Good places to visit.... happy trails:

1. Guessing correlation
2. About correlation
3. NIST handbook
4. Example with formulae
5. Correlation, regression, outlier points
6. Correlation lesson for psychologists
7. With an ellipsoid

Outliers:
NIST handbook