Interpreting r

r tells you two things at once: direction, from the sign, and strength, from how close the value sits to -1 or 1. An r near 0 means points scatter with no clear linear trend. An r near 1 or -1 means the points hug a line tightly. But r = 0.8 does not mean the relationship is '80 percent as strong as a perfect one.' The jump in tightness from r = 0.7 to r = 0.95 is much bigger visually than from 0.3 to 0.7 — the scale is not linear in any intuitive sense. The only quantity with a clean percentage meaning is r-squared, the percent of variation in y explained by the linear relationship with x.

Two things trip people up constantly. First, strong correlation is not causation: two variables can move together because a third variable drives them both, or just by coincidence, so r alone never proves one causes the other. Second, r is not resistant. Because its formula depends on every point's distance from the means in both variables, one far-flung point can drag r up or down dramatically, the same way an outlier distorts the mean more than the median.

This sets up the rest of the unit: r only describes linear relationships, so you always check the scatterplot and the residual plot before trusting it, and you watch how restricting the range of x values shrinks r even when the underlying relationship hasn't changed. Interpreting r correctly is what keeps you from overselling a line that doesn't deserve the praise.