Regression Line Intercept

Every least-squares line passes through the point of averages (x̄, ȳ) — that is a consequence of minimising squared vertical errors, not a coincidence. So once you know the slope, the intercept is forced: slide back from ȳ by b times x̄. With ȳ = 50, b = 2.5 and x̄ = 12, the intercept is 50 − 30 = 20, and the fitted line is ŷ = 20 + 2.5x. Adrien-Marie Legendre published the least-squares recipe in 1805; Gauss claimed he had been using it since 1795 and proved it by predicting where the lost asteroid Ceres would reappear, which it duly did.

The trap is over-reading the intercept. It is the fitted value at x = 0, which is often far outside the data and physically meaningless — a regression of adult weight on height gives an intercept for a person of zero height. It exists to anchor the line, nothing more, and a "significant" intercept usually tells you nothing interesting. When x = 0 really is nonsense, centring the predictor (using x − x̄) makes the intercept equal ȳ and gives it an interpretation worth having.