Correlation and Simple Regression

The one idea: Correlation tells you two variables move together; regression draws the actual line through them and hands you a number — "each unit of X is worth this much Y" — that you can put in a forecast, a budget, or a target.

⚠️ The trap: A regression line fit to five or six data points will always draw something, and it will always look confident on a slide. A line is not evidence the relationship is real, linear, or stable outside the range of X you actually observed — it's the one line that best fits the dots you happened to collect, nothing more.

The correlation lesson established that r captures direction and tightness of a straight-line relationship, and that r² tells you what share of the variation is "explained." What r never gives you is a usable number for the business: if r = 0.9 between ad spend and signups, what do you actually get for the next $10,000? Correlation describes the pattern. Regression describes the rate — and that's the number that goes into a plan.

From a cloud of points to one line

Picture a scatterplot of monthly ad spend against trial signups: as spend rises, signups rise, roughly along a diagonal band. Infinitely many straight lines could be drawn through that band. Regression's job is to pick the one line that fits best, using a specific, defensible rule: minimize the sum of squared vertical distances between each point and the line (the "least squares" criterion). Squaring does two things — it makes every miss count as positive (no cancellation between overshoots and undershoots), and it…