A reliable method
Substitute x into the fitted line to predict y. Calculate residual as observed y minus predicted y. In original units, the least-squares slope is correlation times response SD divided by explanatory SD. When both variables are standardized, the slope equals the correlation.
The common trap
A large absolute correlation does not prove causation. Extrapolation outside the observed explanatory-variable range can fail even when the fitted line works well inside it.
Make the connection
Interpret r squared as the proportion of variation in the observed response explained by the fitted linear relationship. It is not the percentage of predictions that are exactly correct.
Prediction and residual tell different stories
For y-hat=19+2x at x=1, the model predicts 21. An observed y=20 gives residual −1, so the model overpredicts by 1. The residual is a vertical difference at a fixed x, not the distance measured along the fitted line and not predicted-minus-observed.
Slope, correlation and explained variation
With r=.6, response SD 10 and explanatory SD 2, the raw-unit slope is 3. With both variables standardized, the slope is .6. Squaring r gives .36, interpreted as 36% of response variation explained by the fitted linear relationship, not 36% of predictions being exactly correct.
Transfer check
Describe the limits of using a line fitted for x between 1 and 5 to predict at x=20. Identify the extrapolation and explain why a high correlation inside the observed range cannot validate that prediction. Also explain why an observational relationship alone does not establish a causal effect.
A fitted line predicts y-hat = 11 + 2x. What response is predicted at x = 4?
- Substitute x = 4 into the fitted line.
- Compute 11 + 2 × 4.
- The predicted response is 19.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework