Correlation
Two scatter plots both climb from left to right. How can one number say which cloud hugs a straight line more tightly?
Where you are 3 inside
What this area is about
Looking at a scatter plot, people agree quickly on direction and form, but not on strength: one calls a cloud “fairly tight”, another “quite loose”. Correlation settles that with a single number, r, which runs from -1 to 1. Its sign gives the direction of a straight-line pattern and its size gives the strength. It measures only straight-line patterns, and it says nothing about cause.
How these fit together
The definition. The correlation coefficient says what r is, what values near 1, 0 and -1 look like, and that it always lies between -1 and 1.
Results. Properties of the correlation coefficient says what does not move r (changing units, swapping the two variables) and what does: a single unusual point can move it a lot.
The procedure. Correlation from standardized values opens the box: r is the sum of the products of the two variables’ z-scores, divided by n - 1, and that formula shows why it can never pass 1 or -1.
A path through it
Start with the correlation coefficient itself. Then take its properties and the z-score computation, in either order.
With r in hand you can build the least-squares line, whose slope is r times a ratio of spreads, measure fit with r^2, and learn why correlation is not causation.