Extrapolation and Prediction

A least-squares line is built entirely from the points you actually collected. It describes the pattern between the lowest and highest x-values in your data, nothing more. Outside that range, the line is just an equation extending forever in both directions — it keeps going because algebra lets it, not because the real relationship keeps going that way. Nothing in the data told the line what happens out there, so any prediction past the edge is a guess dressed up as a calculation.

This matters because relationships that look straight over a limited window often curve, flatten, or reverse once you leave it. Height versus age is a classic case: from age 2 to 10, growth is roughly linear, so a line fits well and r-squared might be high. But growth slows sharply after puberty and stops in adulthood. A line fit to ages 2-10 has literally zero data points anywhere near age 40, so it cannot know that growth has stopped — it just keeps adding height per year forever, producing an absurd prediction.

A high r-squared tells you the line fits the points you have very well — it says nothing about points you don't have. This connects directly to residual analysis: residuals only exist where you have data, so they can warn you a line is wrong inside the observed range but they are silent about anything beyond it. The lesson carries forward whenever you report a prediction: always check the x-value lies within, or close to, the range of the original data before trusting y-hat.