A level-readyAll subjects · Mathematics · WST01 (S1)
Mathematical models in statistics
A statistical model is a simplified description of a real situation, written in the language of probability. S1 starts here because every later topic (the normal distribution, regression lines, discrete distributions) is a model.
What a model is
A mathematical model describes a real-world situation using equations, probabilities or distributions. It is always a simplification: it keeps the features that matter and ignores the rest. For example, 'each face of a die has probability 1/6' models a real die, which may not be perfectly fair.
The modelling cycle
1. Recognise a real-world problem.
2. Devise a mathematical (statistical) model, stating the assumptions made.
3. Use the model to make predictions.
4. Collect experimental data.
5. Compare the observed data with the predictions.
6. If the agreement is poor, refine the model and repeat from step 3.
| Stage | Die example |
|---|---|
| Model | Assume the die is fair: P(6) = 1/6 |
| Predict | In 300 throws expect 50 sixes |
| Collect data | Throw it 300 times; record 71 sixes |
| Compare | 71 is a long way from 50 |
| Refine | Drop the fairness assumption; estimate P(6) from data |
Advantages and limitations
- Advantages: a model is quick and cheap to use, lets you make predictions, helps you understand the situation and can be improved as data arrive.
- Limitations: it simplifies reality, so predictions are never exact, and it only works within the conditions it was built for.
In exam answers, link your comment to the context in the question.
Worked example
A spinner has 4 equal sectors. A model says each sector is equally likely. Predict how many times sector A comes up in 200 spins, and say what you would do if it came up 81 times.
- Model: P(A) = 1/4.
- Prediction: 200 × 1/4 = 50.
- 81 is far from 50, so compare and refine: the spinner may be biased; estimate P(A) from the data, 81/200.
Answer: Predict 50; with 81 observed, refine the model (for example P(A) ≈ 0.405).
Check yourself
- A model assumes a die is fair. How many sixes does it predict in 300 throws?
Show answer
50. 300 × 1/6 = 50.
- A model assumes boys and girls are equally likely at birth. How many boys does it predict in 240 births?
Show answer
120. 240 × 0.5 = 120.
- In the modelling cycle, which stage comes immediately after 'use the model to make predictions'?
- Refine the model
- Recognise the real-world problem
- State the assumptions
- Collect experimental data
Show answer
Collect experimental data. You need real data before you can compare it with the predictions.
- Which is an advantage of using a statistical model?
- It is a quick and cheap way to make predictions
- It always gives exact predictions
- It removes the need to collect data
- It includes every feature of the real situation
Show answer
It is a quick and cheap way to make predictions. Models are cheap and fast, which is why we use them.
- Which is a limitation of a statistical model?
- It cannot be changed once written
- It can only be used for dice
- It simplifies reality, so its predictions may be inaccurate
- It needs a computer to use
Show answer
It simplifies reality, so its predictions may be inaccurate. Simplification is the price of a manageable model.
- A model predicts 50 sixes in 300 throws, but 71 are observed. What is the best next step?
- Refine the model, for example by not assuming the die is fair
- Accept the model, because 71 is close enough to 50
- Throw the die 3 more times
- Discard all the data
Show answer
Refine the model, for example by not assuming the die is fair. Poor agreement between data and prediction means the model needs refining.
- A die is thrown 300 times and shows 71 sixes. Using the data, estimate P(six) for a refined model. Give your answer to 3 decimal places.
Show answer
0.237. 71/300 = 0.237.
Sources
Last updated 7 November 2026. Spotted a mistake? Tell us.