ACT® · Math: The Modeling Overlay
Evaluating Whether a Model Is Good Enough
A good model doesn't need to hit every point exactly. It needs small, patternless errors, and you check that by looking at the shape of the residuals, not just their size.
A model is a simplified rule standing in for messy real data. No simple rule hits every point, so the question is never "is it perfect" but "is it close enough, and close in the right way." A model that misses by a little everywhere is usually more useful than one that nails two points exactly and is wildly off elsewhere, because the second one has no consistent relationship to the data between those points. You cannot trust it anywhere except the two spots you already checked.
The size of the errors tells you how far off you are. The pattern of the errors tells you whether you picked the right kind of model at all. Residuals are actual minus predicted. If you line them up in order of x and they look random, scattered above and below zero with no trend, the model family is probably right. If instead they march steadily from negative to positive, or dip down then up, that shape means the real data curves in a way your model's shape cannot follow, even if every single residual is small. A straight line can leave small but systematically patterned residuals when the truth is a curve.
This connects directly to percent error, which measures how big a single miss is relative to the actual value: (predicted minus actual)/actual, times 100. Percent error judges one prediction; residual patterns judge the whole model across many points. Together they let you both quantify a model's error and diagnose why it's happening, which matters once you start adjusting parameters or switching model families to fit the data better.
Key terms
- residual
- The actual value minus the predicted value at a given point.
- percent error
- The size of a prediction's miss divided by the actual value, times 100.
- model family
- The general shape of rule used to fit data, such as linear, quadratic, or exponential.
- systematic pattern
- A trend in residuals, like steadily increasing, that signals the model shape is wrong even when errors are small.
Watch it explained
- How it worksHow does the pattern of the errors, rather than their size, expose the wrong model family?
- Worked exampleFind whether a linear model is acceptable when its errors run -4, -1, 2, 5 across four points.
- Worked exampleFind the percent error when a model predicts 480 and the actual count comes in at 512.
- Going furtherWhat happens to your verdict on fit when the measurements themselves carry error?
Check yourself
End of topic.