Mistake Master
The Line That Loses Least, and the Number Everyone Misreads
You'll learnwhat the least-squares criterion actually minimizes — squared, vertical, minimized, three words each hiding a standard wrong answer — how the slope comes from r and the two standard deviations, why the line always passes through the point of averages, what r-squared counts and what it does not, and how one high-leverage point can reverse a slope.
Eight students report their weekly study hours and their exam scores, and of all the lines that could be drawn through that cloud, least squares picks exactly one: score = 58.5 + 3.5 · hours. The rule it uses is more specific than it sounds — it makes the sum of the SQUARED VERTICAL residuals as small as possible, and this page runs 5,751 rival lines to show that none of them beats 25.0. Then the number everyone misreads: r-squared = 0.954 says the linear model with study hours accounts for 95.4 percent of the variability in scores. Read as "95.4 percent of the points are on the line" it predicts 7.6 of the 8 points; the actual count is 1. And the whole fit hangs on every point being ordinary: add one student at 20 hours and a score of 40, and the slope flips from +3.5 to −1.27.