Pearson's r runs from minus one through zero to plus one. Visualise scatterplots at each value.
From the chapter: Chapter 4: Probability
Glossary: correlation, covariance
Transcript
Two variables, X and Y. We want a number that summarises how they move together.
The covariance is the average of X minus its mean times Y minus its mean. Positive when they move together, negative when one rises as the other falls.
But covariance has units. Larger-scale variables produce larger covariance. To compare across pairs, divide by the product of standard deviations. The result is Pearson's correlation coefficient, capped between minus one and plus one.
When r equals plus one, the relationship is perfectly positive. Every data point sits exactly on a line that rises to the right.
At r equals zero, no linear pattern remains. The cloud may be round, or it may have a non-linear shape that this measure cannot see.
With r equals minus one, the relationship is perfectly negative. The points fall exactly along a line that drops to the right.
In between, more spread.
Crucial caveats. r is purely linear. A clean parabola has correlation zero. A non-linear monotonic shape has correlation that depends on the curvature.
And correlation is not causation. Two ice cream variables can be correlated through a hidden cause, summer. Always look at the scatter plot, not just the number.