What Correlation Measures

A scatterplot full of dots is hard to summarize in words, but if the dots roughly trace a line, one number can capture how tightly they hug it and which way it tilts. That number is r. To build it, turn every x and every y into a z-score, so each measures how many standard deviations a value sits above or below its own mean. Multiply each point's z_x by its z_y, then average those products across all the points. That average is r.

Think about what the sign of each product tells you. A point above-average in both x and y, or below-average in both, gives a positive product, since two negatives multiply to a positive. A point that's above-average in one and below in the other gives a negative product. If most points fall in the quadrants where x and y move together, positive products dominate and r comes out positive and close to 1. If points scatter evenly across all four quadrants, the positive and negative products cancel, and r lands near 0. Points far from both means, in standardized units, contribute the most weight, which is why outliers can swing r a lot.

Correlation only ever measures straight-line association. That restriction is exactly why the next idea, the least-squares regression line, exists: r tells you how well a line fits, and the line itself gives you the equation for predicting y from x. Residuals later check whether that linear story was ever the right one to tell.