S1.1 - Mathematical models in probability and statistics
- Syllabus
- 2019
- Topic
- S1.1
- Level
- AS
Learning objectives
Use and refine a statistical model
A mathematical model is a simplified representation of a real situation. In probability and statistics, it keeps the variables and relationships needed for a purpose while replacing some real-world complexity with stated assumptions.
| Stage | Modelling decision |
|---|---|
| Define | State the real question, the variables or outcomes of interest, and what the model should help explain or estimate. |
| Simplify | Choose assumptions and omit details judged not to matter for this purpose. |
| Use | Analyse the model to reveal a relationship or trend, solve a real-world problem, or improve understanding. |
| Check | Interpret the result in context and compare the model's behaviour with relevant observations. |
| Refine | Adapt assumptions, variables or structure when the model is not adequate; the revised model can be used and checked again. |
A model is useful because it makes a complicated situation manageable and repeatable. The same structure can be applied to new data, relationships between variables can be made clearer, and assumptions can be changed to investigate whether a conclusion is robust.
Suppose a school models student journey times using ordinary weekday travel records. The model may deliberately ignore rare road closures and assume future weekdays resemble the recorded ones. It can help describe the usual pattern and plan an arrival time. If later observations consistently disagree, the school should reconsider the data period or assumptions rather than force the situation to fit the old model.
A model is not the real world and is not automatically reliable. Its conclusion is conditional on its assumptions, selected variables and data. Simplification is useful only when omitted factors do not materially undermine the model's stated purpose; agreement in one setting does not guarantee valid use in another.