Slope, Intercept, and the Point of Means

The least-squares line is built to balance positive and negative residuals above and below it, and that balancing point is always (x-bar, y-bar). No matter how scattered the data are, the line of best fit runs through the spot where both variables sit at their average. That single fact is what lets you find the equation of the line without touching calculus: once you know the slope, the point of means pins down the intercept for free.

The slope formula itself comes from stretching correlation back into the original units. Correlation r measures how tightly points cluster around a line, but it's unitless — it doesn't know whether x is measured in inches or in miles. The slope needs to know that, because slope is rise over run in the real units of y and x. Multiplying r by sy/sx rescales the unitless strength of the relationship into an actual rate of change: slope b = r(sy/sx). Once b is known, intercept a = y-bar − b(x-bar) follows directly from the line passing through the point of means.

This formula is also why slope and correlation get confused but aren't the same thing. They connect Unit 5's earlier ideas — correlation measuring strength and direction — to prediction: slope tells you how much y changes per unit of x, while r only tells you how closely the points hug a line. Next, residuals check whether that line was ever the right model to use.