Evaluating and Extending a Regression Model

The one idea: R² tells you what share of the variation your line explains, and residual analysis tells you whether the line was even the right shape to fit in the first place — you need both, because a high R² with a broken residual plot is a well-fit wrong model.

⚠️ The trap: "Our model explains 94% of the variance, so it's ready to drive decisions." R² measures fit to the data you already have — it says nothing about whether the relationship is causal, whether it holds outside that data, or whether the model's assumptions are even approximately true. Four datasets can share the exact same R² and look nothing alike once you plot the residuals.

The previous lesson fit a line — ŷ = 3.7 + 2.1x — to five months of ad spend and signups. A fitted line always exists; least squares will hand you one for any set of points, including ones with no real relationship at all. This lesson covers how to grade the line you got: how much of the variation it actually accounts for (R²), how much typical error to expect on any one prediction (standard error of the estimate), how to check the assumptions behind both numbers (residual analysis), and what changes when you add a second predictor (multiple regression).

Splitting the variation: SST, SSR, SSE

Every regression evaluation starts from one accounting identity. Take the ad-spend example's five signup values: 6, 8, 9, 13, 14, with mean ȳ = 10.

Total variation (SST) — how spread out y is if you ignore x entirely and just use the mean as…