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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WST01 · Statistics 1Mathematical models in probability and statisticsStatistical models and the modelling cycle

Mathematical models in probability and statistics · Subtopic 1 of 1

Statistical Models and the Modelling Cycle

A statistical model is a simplified mathematical description of something random in the real world. You design it, use it to predict, collect real data to test it, and refine it when predictions and data disagree. Models are quick and cheap but never perfect.

Edexcel YMA01 WST01≈ 20 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Explain what a statistical model is.
Describe the stages of the modelling cycle in order.
Give advantages and disadvantages of models.
Decide when data suggest a model needs refining.

Pause and think

A weather app says " $30\%$ chance of rain" and it rains. Was the model wrong?

Section 2 of 14

Core idea

Start hereA map is not the territory

A tube map ignores real distances, yet it's perfect for planning a journey. A statistical model is the same: a deliberate simplification of reality that's useful as long as you remember what it leaves out.

The big idea

Assume, choose a distribution, predict, compare with data, refine. Models are quick and cheap but only as good as their assumptions.

1. The cycle

The modelling cycle

Make assumptions, build the model, use it to predict, test against data, then improve the assumptions and go round again.

Real-world problem → assumptions.
Choose a model (binomial, normal, ...).
Predict.
Collect data, compare, refine.

Worked exampleModelling heights

Model adult women's heights and test the model.

Goal: A working model.
Strategy: Follow the cycle.

Assume heights are symmetric and bell-shaped.
$H\sim N(165,49)$
Choose parameters from a pilot sample.
$P(158<H<172)=P(-1<Z<1)=0.683$
Prediction: about 68% within 7 cm.
Compare with a real sample; if the tails don't fit, adjust the mean or variance.

AnswerA normal model, tested by comparing predicted and observed proportions.

✓ CheckHeights can't be negative, but

165

is 23 SDs above

0

: no practical issue ✓.

2. Advantages and limitations

Advantages: quick, cheap, allow prediction and understanding.
Limitations: simplified; assumptions may fail; predictions only as good as the model.

Common misconceptionModel = reality A model that fits past data may still fail in new conditions.

Pause and think

Why might a normal model be poor for incomes?

Summary

Assume, model, predict, test, refine.

Model: Simplified mathematical description.
Assumption: A simplification accepted to build a model.

History"All models are wrong…" The statistician George Box wrote in the 1970s that "all models are wrong, but some are useful". It became the motto of applied statistics: the question is never whether a model is perfect, but whether it's good enough for its purpose.

Fun factForecasting elections. Election forecasters build models from polls, past results and population data, then refine them after every election. Each surprise result is "comparing predictions with data", the most painful step of the cycle.

Section 3 of 14

Key formulae

The cycle

\text{observe}\to\text{model}\to\text{predict}\to\text{collect data}\to\text{compare}\to\text{refine}

TrickMore data, fairer test. Judge a model on lots of data.

2

sixes in

6

rolls proves nothing;

190

sixes in

600

rolls (expected

100

) is strong evidence the "fair dice" model is wrong.

Section 4 of 14

Worked examples

Example 1 · Advantage and disadvantage0 / 1 steps

Give one advantage and one disadvantage of using a statistical model to predict crop yields.

Example 2 · Refining0 / 1 steps

A model predicts

30

120

seeds fail;

58

fail in an experiment. What should happen next?

Section 5 of 14

Explore the graph

Expected and observed sixes

Loading the interactive graph…

Try this

The line is the "fair dice" model: expected sixes $=\tfrac16$ of the rolls. The upper point is an observed count $o$ after $600$ rolls; the lower is the expected $100$ .
How far from $100$ would $o$ need to be before you'd refine the model?
Would the same gap matter as much after only $60$ rolls?

Compare predictions with data, and refine when they disagree clearly.

In the real worldInsurance prices. Insurers model how likely drivers are to claim. When claims data show a group behaving differently from the model, they refine it, which is why prices change with age, area and experience.

Section 6 of 14

Common mistakes

Incorrect"Models are exact"

CorrectThey are simplifications

Something is always left out.

IncorrectRefining after a tiny sample

CorrectUse plenty of data

Small samples vary a lot by chance.

IncorrectCollecting data before designing the model

CorrectModel first, then test with data

Follow the cycle's order.

IncorrectVague disadvantages

CorrectName what is ignored

"It's inaccurate" scores nothing.

The model is reality
The model is wrong, so it's useless
All models are wrong; some are useful. Refine and repeat.

Section 7 of 14

Exam tip

Exam tipAnswer in context. Modelling questions reward context. Instead of "it may not be accurate", say "it ignores the effect of the season on rainfall". One specific sentence earns the mark.

Section 8 of 14

Guided practice

SA-IAL-S1-MD-Q011 markL1 FoundationCalculatorModelling cycle

A statistical model is

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-S1-MD-Q021 markL1 FoundationCalculatorModelling cycle

Which is the correct order of the modelling cycle?

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q061 markL1 FoundationCalculatorModelling cycle

In the modelling cycle, data are collected in order to

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SA-IAL-S1-MD-Q101 markL1 FoundationCalculatorModelling cycle

After a model has been refined, the next step is to

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q031 markL1 FoundationCalculatorEvaluating models

An advantage of using a statistical model is that it

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q041 markL1 FoundationCalculatorEvaluating models

A disadvantage of a statistical model is that

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q081 markL1 FoundationCalculatorEvaluating models

Which is NOT a reason for using a model?

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q111 markL1 FoundationCalculatorEvaluating models

Making a model simpler usually makes it

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q051 markL2 StandardCalculatorEvaluating models

A model predicts that 30 out of 120 seeds will fail to grow. In an experiment, 58 fail. The most sensible next step is to

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-MD-Q091 markL2 StandardCalculatorEvaluating models

The heights of adult men are modelled by a normal distribution. A limitation of this model is that

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-S1-MD-Q121 markL2 StandardCalculatorEvaluating models

A model of daily rainfall is based on one month of data. Why might its predictions be poor for the whole year?

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

A model is a simplified description used to predict.
Cycle: observe, model, predict, collect, compare, refine, repeat.
Pros: quick, cheap, insightful. Cons: simplified, approximate.
Refine when lots of data disagree clearly.

Section 12 of 14

Quick quiz

Q1 · SA-IAL-S1-MD-Q071 markL2 StandardCalculatorEvaluating models

A fair die is modelled with P(each score) = 1/6. Which observation would most suggest the model needs refining?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-S1-MD1-0011 markL1 FoundationCalculatorEvaluating models

A model predicts

25

heads in

50

tosses of a coin. An experiment gives

27

heads. The most sensible conclusion is that

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-IAL-S1-MD1-0021 markL1 FoundationCalculatorModelling cycle

In the modelling cycle, what comes immediately after "use the model to make predictions"?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WST01 (Statistics 1)
Where: S1 · 1 Mathematical models in probability and statistics
Expected: Describe the process of mathematical modelling: observe, design a model, predict, collect data, compare and refine; state advantages and disadvantages of models.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: Short written answers: "Give one advantage and one disadvantage of using a statistical model."
Open in: YMA01 · WST01

Section 14 of 14

Past paper questions

1 question (14 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WST01/01 · June 2025Question 314 marks
Modelling, correlation and regression from coded summary data.
Question paper, page 6 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).