Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- The least-squares regression line ŷ = a + bx is the one line with the smallest sum of squared residuals, Σ(y − ŷ)².
- You find a, b, r and r² with technology. Enter the explanatory variable as x.
- The line always passes through (x̄, ȳ), and its residuals add to 0.
- a and b are statistics from one sample; a different sample would give different values.
Key relationships
| Quantity | Meaning | Interpretation template |
|---|---|---|
| Slope b | predicted change in y per 1-unit increase in x | “For each additional [unit of x], the predicted [y] increases/decreases by [b units].” |
| Intercept a | predicted y when x = 0 | “When [x] is 0, the predicted [y] is [a units].” Say if it has no sensible meaning. |
| r | direction and strength of the linear association | same sign as b; −1 ≤ r ≤ 1 |
| r² | coefficient of determination, 0 ≤ r² ≤ 1 | “About [r²%] of the variation in [y] is explained by the linear relationship with [x].” |
| (x̄, ȳ) | always on the line | ȳ = a + b·x̄, so a = ȳ − b·x̄ |
| Background | b = r·s_y / s_x | not needed for calculation; explains why b and r share a sign |
Assumptions and conventions
- Only fit a line when the scatterplot looks linear and the residual plot shows no pattern.
- The intercept has no reasonable meaning when x = 0 is outside the data (extrapolation) or the prediction is impossible (for example a negative length).
- In technology output, “Constant” is a; the row named after x is b; “R-sq” is r².
Mistakes to avoid
- Leaving out “predicted” when interpreting the slope or intercept.
- Interpreting an intercept that makes no sense instead of saying so.
- Calling r² “the percentage of points on the line” or “how often the line is right”.
- Taking r = +√r² when the slope is negative.
- Swapping x and y in technology: the line changes.
- Treating a steep slope as a strong association: strength comes from r, not b.
- Claiming cause and effect from a regression line.
Quick self-check
- r² = 0.49 and the slope is negative. What is r? (r = −0.7)
- An LSRL has slope 2, x̄ = 12 and ȳ = 50. What is the intercept? (a = 50 − 2(12) = 26)
- ŷ = 3.1 + 0.42x. What is the predicted y at x = 20? (3.1 + 8.4 = 11.5)
- ŷ = −40 + 0.9x predicts a mass (g) from a length (mm) for lengths 80 to 150 mm. Should you interpret the intercept? (No: x = 0 is outside the data and −40 g is impossible.)
Next: practice questions.