The big idea: Linear regression predicts a continuous number by fitting a straight line through the data.
The model is just — and training means finding the and that make the line fit best.
What makes it regression rather than classification
- The output is a number on a scale — price, temperature, minutes, marks
- Not a category — spam or not, cat or dog, pass or fail
- "How much?" is regression · "which one?" is classification
- The same input data can support either, depending on what you ask
The words the guide uses
| Term | Meaning | House-price example |
|---|---|---|
| Independent (predictor) variable, x | The input used to make the prediction | Floor area in m² |
| Dependent (response) variable, y | The continuous value being predicted | Price |
| Slope, m | How much y changes when x goes up by 1 | Price per extra m² |
| Intercept, c | The value of y when x is 0: where the line crosses the y-axis | Often has no real meaning (a 0 m² house) |
| r² | How well the line fits: the share of the variation in y the model explains, from 0 (none) to 1 (all) | r² = 0.85: the line explains 85% of the variation in price |
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Measure the error
For each point, the residual is the gap between the real value and the line's prediction.
Square it
Squaring makes every error positive, so errors above and below cannot cancel out — and it punishes large errors much more than small ones.
Minimise the total
The best line is the one with the smallest sum of squared residuals. This is why the method is called least squares.
Read off the parameters
The resulting m (gradient) says how much y changes per unit of x; c (intercept) is the prediction when x is zero.
The gradient is the interpretation: In a model predicting house price from floor area, m is the price per square metre.
That is why regression is prized where a decision has to be explained: the parameter itself is a plain-English statement about the world.
Squaring is why outliers hurt: An error of 10 contributes 100; an error of 100 contributes 10,000. One wildly wrong point can drag the whole line towards itself, which is why outliers matter so much more here than in some other methods.
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| Works | Fails | |
|---|---|---|
| Relationship | Roughly straight | Curved, or a threshold effect |
| Output | Continuous | Categories |
| Outliers | Few | Any — squaring amplifies them |
| Extrapolation | Inside the observed range | Outside it — the line keeps going, reality may not |
Extrapolation is the classic failure: A line fitted to house sizes between 50 and 200 m² will happily predict a price for 5,000 m², or a negative price for 0 m².
The model has no concept of the range it was trained on. Predicting outside it is the commonest way regression gives an absurd answer with full confidence.
Correlation is not causation: A regression showing ice-cream sales predicting drownings has found a real correlation with a third cause — hot weather. The model is not wrong; the causal reading of it is.
How this is tested — you must decide whether the target is continuous, interpret the gradient, and know where the line stops being trustworthy. It comes up two ways:
Paper 1 Section A
- Explain how linear regression predicts, 3-4 marks
- Identify whether a task is regression or classification
- State what the gradient means
Paper 1 Section B — case study
- Interpret a fitted model in context
- Explain why a prediction is unsafe
The classic trap: Reading a gradient as a cause. Regression finds that two things move together; it says nothing about one producing the other. Say "associated with", not "causes".
An estate agent fits price = 2400 × area + 18000, using homes between 50 and 200 m². Explain what the model says, and why its prediction for a 600 m² property should not be trusted.
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