Module 6 · Regression & Correlation · Topic 5
Least Squares. The least squares method finds the line that minimizes the sum of the squared residuals (SSE = Σ(yᵢ − ŷᵢ)²). This produces the best-fitting line in terms of vertical distances from data points.
Why Squared?. Squaring the residuals ensures positive and negative errors don’t cancel, penalizes larger errors more, and yields a unique, mathematically tractable solution.
b = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / Σ[(xᵢ − x̄)²]
yhat = a + bx — Line that minimizes the sum of squared residuals.r = sum((x_i-xbar)(y_i-ybar)) / sqrt(sum((x_i-xbar)^2)sum((y_i-ybar)^2)) — Measures direction and strength of a linear association.Imagine pinning a thread to a scatter plot so it passes through (̄x, ȳ). Rotating it until the sum of squared vertical residuals is minimized gives the least squares line.
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