Worked example 1: A sensor calibration dataset clusters around a straight line, but one endpoint sits far above it. What should you inspect first?

plot residuals, not only the fitted line
A visually reasonable line can still hide a systematic failure at one edge of the data range.
one endpoint residual is unusually large
That may mean an outlier, sensor saturation, or the beginning of curvature the linear family cannot capture.
do not trust extrapolation past that region
A calibration fit is only as trustworthy as the range in which the residuals stay defensible.

Worked example 2: A bacterial culture has counts 100, 200, 400, 800 at equal time steps. Should the first model be linear or exponential?

differences: 100, 200, 400
The added amount is not constant, so a linear family is already under pressure.
ratios: 2, 2, 2
Equal multiplicative steps strongly support an exponential first model.
use exponential until evidence of saturation appears
If later data bend toward a ceiling, the model should be upgraded to logistic instead of forcing exponential forever.

Worked example 3: A population rises quickly, then the yearly increases shrink as the count approaches a carrying capacity. What is the warning against overfitting here?

a high-degree polynomial can wiggle through every point
A very flexible curve can memorize the sample without reflecting the mechanism.
logistic family is mechanistically cleaner
It encodes the ceiling directly instead of hiding it inside arbitrary oscillations.
prefer interpretable parameters and stable residuals
Good models generalize. Overfit models often produce brittle extrapolations and meaningless parameter stories.

Common mistake: treating the curve that passes closest to every point as automatically best. A model can be accurate on the sample and still be scientifically poor.

Worked example 4: A shipping-cost dataset is flat below a weight threshold and then rises linearly. Why is a piecewise model better than one smooth polynomial fit?

there is a real threshold in the mechanism
The pricing rule changes at one known point, so the mathematics should preserve that regime change.
piecewise model keeps the rule interpretable
A single polynomial may smooth over the threshold and invent behavior that the pricing system never actually uses.

Functions