Pearson's product-moment correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two numerical variables x and y. Its value always lies between −1 and +1:

• r=+1: perfect positive linear relationship

• r=−1: perfect negative linear relationship

• r=0: no linear relationship

A positive value means y generally increases as x increases; a negative value means y generally decreases as x increases. The closer r is to ±1, the stronger the linear relationship.

Pearson’s coefficient is calculated by:

Note: you do not need to use or know this formula.

On exams, r is determined using technology.

Example 1

Find r for the following data, and interpret its significance.

Using a calculator, we find r=−0.973. This indicates a very strong, negative relationship.

Example 2

Find r for the following data, and interpret its significance.

Using a calculator, we find r=0.11. This indicates essentially a weak positive relationship, but there is almost no correlation.