Definition of the correlation coefficient
The strength of a relationship is often summarised by a correlation coefficient. Its usual formula is complex but it can be expressed more simply in terms of z-scores.
How does r relate to the shape of a scatterplot?
Although it is difficult to prove some of these properties, it is possible to get a feel for how the value of r reflects the strength of a relationship from its definition. The diagram below explains.
Strength of the relationship and r