Correlation vs Association

Correlation reports how tightly the points in a scatterplot hug a straight line and which way that line tilts, and it carries no units, so switching from inches to centimeters leaves it unchanged. The value $r$ is computed as $r = \frac{1}{n-1}\sum \left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)$, where $\bar{x}$ (read x-bar) and $s_x$ are the mean and standard deviation of $x$. The catch is that $r$ sees straight-line patterns only: the five points $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 4)$ follow $y = x^2$ perfectly, yet $r = 0$. Correlation is also not resistant, so one far-out point can drag $r$ up or down on its own.

Association is the broad idea and correlation is one narrow measure of it. For two quantitative variables you describe an association by its form (linear or non-linear), direction (positive or negative), strength, and unusual features, so a curved pattern still counts as a real association. For example, braking distance grows roughly with the square of speed, so 20, 40, and 60 mph pair with about 20, 80, and 180 feet: doubling the speed quadruples the distance ($4 \times 20 = 80$), a strong association that no straight line captures well. Association also applies to two categorical variables, where you compare conditional percentages in a two-way table and correlation does not apply at all.