How big should a school poll be? Sampling from a known population

AI SLAI HLAA SL Solid Also in: Simulation, Statistics

Research question

By taking repeated samples of different sizes from a dataset where the true answer is known, how does the margin of error shrink with sample size, and how many students would a school poll need for a margin of ±5 percentage points?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

It makes sampling error visible: you see the spread of estimates around a known truth before you trust your own single poll.

The mathematics you'll need

Random sampling methods
Sampling distribution of a proportion
Standard error √(p(1 − p)/n)
Margin of error and sample size
Comparing random, convenience and stratified samples

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use a public dataset as a 'population' (the UCI repository has a student-performance dataset), then run your own small poll.

UCI Machine Learning Repository — 600+ classic, documented datasets (wine quality, student performance, bike sharing…).
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Choose a population dataset and a yes/no variable.
Take many samples of each size and record the estimates.
Compare the spread with the standard-error formula.
Compare sampling methods.
Plan and run your own poll and reflect.

Pitfalls that cost marks

Too few repeated samples.
Treating a convenience sample as random.
Using the formula without checking it against the simulation.

Showing personal engagement

Run a real poll on a school question.
Predict the needed sample size first.
Test a convenience sample against a random one.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Random sampling methods; Sampling distribution of a proportion
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Random sampling methods; Sampling distribution of a proportion
AI HL	Good fit	Random sampling methods; Sampling distribution of a proportion

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Design every part yourself: the question, the sampling frame, the wording and the analysis plan. Piloting the questionnaire and changing it is engagement an examiner can see.

Reflection (D)

Reflect on bias (who answered, who didn't, how wording steered answers) and on what a significant result can and can't show about the whole population. For this idea, start with: too few repeated samples — say how it affects your answer.

Use of mathematics (E)

SL: A sampling method justified, a sample size worked out from the test you plan, the test's conditions checked, one calculation shown and the p-value interpreted in context.

HL: Confidence intervals, a justified choice between tests, a test of a model's fit, or a statistical estimate of how much bias could change the conclusion.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Adjust for the finite population of your school and see how much it matters.

Extending it for HL

Add confidence intervals for the proportions you report, and estimate how large a non-response bias would have to be to overturn your result.

See a complete IA, marked

Our annotated exemplar Do students who sleep less react more slowly? (AI SL) asks a different question, but shows how a complete surveys exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).