Scatter Plots and Correlation

Correlation is simply a measure of how closely two things move together in a straight-line way, like study hours and exam scores. The formula looks intimidating, but it's built from pieces you know: $\frac{x - \bar x}{s_x}$ (the z-score of $x$, how far above or below average it is) times $\frac{y - \bar y}{s_y}$ (the z-score of $y$), added up over all the points, then divided by $n - 1$ (roughly an average). So $r$ is basically the average product of z-scores. Bear with it; the next paragraph shows why that's such a natural thing to compute.

Reading $r$: think about what that product does. If a point is above average in both $x$ and $y$, both z-scores are positive and the product is positive. If it's below average in both, two negatives make a positive again. Those are the upper-right and lower-left quadrants around the means, and they push $r$ up. A point that's high in one and low in the other gives a negative product and pulls $r$ down. So $r$ ends up positive when the points mostly sit in the "both high or both low" quadrants, and negative when they don't.

Properties:

• $r$ has no units, and changing units (inches to centimeters) doesn't change it. That's because it's built from z-scores, which have already divided the units out.
• $r$ only measures linear association. A perfect parabola can have $r = 0$.
• $r$ is sensitive to outliers.

A common mistake. When two variables are strongly correlated, it's tempting to conclude one causes the other. Correlation is not causation. Two variables can move together because a lurking variable drives both (bigger fires bring more firefighters and more damage), or by plain coincidence.

Finance tracks correlations between assets to diversify, medicine screens for risk factors, and data scientists check correlations before building models.