Key Idea: When data curve, a straight line is the wrong model. You choose a curved model, find its constants by regression, judge it with R², and use it to predict.
Look, fit, judge, predict
- Look — Choose a family that matches the shape of the scatter.
- Fit — Fit it on the GDC. Read the constants and R².
- Judge — R² close to 1 = strong. Compare R² to choose between models.
- Predict — Substitute x, interpret in context, and say if it is reliable.
| Shape | Model |
|---|---|
| Constant % change | Exponential y = k·aˣ |
| Steep then flattens, or scaling | Power y = a·xᵇ |
| Rise then fall | Quadratic |
| S-wiggle | Cubic |
| Repeats over time | Sinusoidal a·sin(bx + c) + d |
A good R² does not make a far-off prediction safe.
P = 8.0·(1.06)ⁿ, n years after 2020. Estimate P in 2030. 8.0 × 1.06¹⁰ ≈ 14.3 thousand.
Is the model safe for 2070? No. n = 50 is far outside the data (extrapolation), and the 6% rate may change.
Model A has R² = 0.91 and model B has R² = 0.99. Which fits better? B.
Wing area against body length rises ever more steeply. Which family? Power.