Maths IA · Statistics and probability · topic 22 of 40

Polls: how big a sample shrinks the margin of error? · IB Maths IA idea: guide, sample graph and step-by-step plan

nice, you can CHECK your own answer

A poll of 1,000 people claims to speak for millions. You're going to find out exactly how much confidence that deserves, and where it runs out.

Difficulty●●○ Medium
KitSpreadsheet or code for simulation
TimeAbout 12 to 15 hours
FitsAI HL, AA HL
  • ✓ you wonder how pollsters get away with 1,000 people
  • ✓ you like simulation
  • ✓ you can compare formula with experiment

The big idea

√n is the key idea

A sample proportion varies from sample to sample. Its spread follows a pattern, and that pattern gives the margin of error. Your exploration derives how the margin shrinks as the sample grows, and tests the claim four times the sample, half the margin. Confidence intervals for a proportion are in neither course (AI HL has them for a mean), so build that step carefully. You test this with simulation on a population you define, and compare the formula with the simulated spread. The best work also asks when the formula fails: small samples, proportions near 0 or 1, or a biased way of sampling that no sample size can fix. At HL, justify the sampling distribution rather than quoting it.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Simulate your own population

Create a population with a known proportion, draw thousands of samples at several sizes and record the spread.

→ You compare an experimental margin of error with the formula for each n, so the formula is tested, not trusted.
Vary the true proportion

Repeat with p near 0.5, 0.2 and 0.05.

→ Your graph shows how the margin depends on p, and you see where the usual approximation struggles.
Real polls

Take published poll results with their reported sample sizes and margins and compare them with your formula.

→ Your model meets real reports; you must discuss why published margins might differ (weighting, design).
Bias versus size

Build a biased sampling rule and show a huge sample still misses the true value.

→ The conclusion moves from 'bigger is better' to 'bigger fixes only random error', a strong reflection point.
Finite populations

Sample a large fraction of a small population, such as your school.

→ You meet the finite-population correction and can derive or test why the margin shrinks faster than usual.
Derive it for HL

Show the variance of the sample proportion from the binomial, then link to the normal approximation and confidence interval.

→ This brings the sophistication and rigour HL criterion E needs; quoting a formula does not.

Your Maths IA research question: run the check

Write your own question first. Then make sure it ticks every box:

  • States what is being estimated (a proportion, with a stated true value in your simulation or a real poll).
  • Says what 'small' margin means in numbers, such as a target width.
  • Says how you will compute the margin: formula, simulation, or both and a comparison.
  • Names the range of sample sizes you will examine and why.
  • Does not rely on a real unknown population: you must be able to check your answer.

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The picture to draw

Draw the population, one sample drawn from it, and the spread of sample proportions when you repeat the sampling. The margin of error is the shaded width.

nsample proportion p̂population: true proportion p is unknownrepeat many times (simulation):this is the spread of p̂how big must n beto shrink the shaded part?shaded: middle 95%

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why polls matter, which poll or election prompted you, what you expect before calculating.
Sampling and the sample proportion~2 pages
Population, sample, random variable p̂, and what 'random sample' means in practice.
The sampling distribution~3 pages
Mean and variance of p̂ from the binomial, the normal approximation and its conditions. At HL, derive and justify.
Confidence interval and margin of error~2 pages
How the interval is built, what '95%' means (and does not mean), and how the margin depends on n and p.
Simulation~2 pages
How the population and samples were built, number of repetitions, a table of simulated spread against n.
Formula versus simulation~2 pages
A graph of both on one set of axes, the gap between them in numbers, and the cases where they differ.
Conclusion~1 page
A rule for choosing n for a target margin, with limits stated.
Reflection and limits~1.5 pages
Bias, non-response, the meaning of 95%, simulation randomness, and one next step with a reason.
Bibliographynot counted
Sources for the theory and any polls quoted.

What your Maths IA graph could look like

state n, p and repetitions

Made-up simulation results, to show the form of a strong graph. example only

025050075010001250150017502000024681012Sample size n (people)95% margin of error / percentage pointsformula, true p = 0.5formula, true p = 0.1simulated half-width of middle 95% (made up)doubling n again and again:what does the curve's shape sayabout the cost of accuracy?ILLUSTRATIVE · MADE-UP DATA
  • Axis titles state 'percentage points', not just '%'.
  • Formula curves for two values of p show that the worst case is p = 0.5.
  • Simulated points hug the curve, which is the evidence that the formula is sound for these n.
  • Start the n axis at the smallest n you tested, and say why.
  • Another graph to consider: the margin against 1/√n, which should be a straight line through the origin.

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Pick a questionA poll or a real count that interests you. Decide what the 'true' value will be in your simulation so you can check your answers.
  2. PredictWrite down how large a sample you think you need for a margin of 3 percentage points. Keep it to compare later.
  3. Learn the mathsBinomial variance, normal approximation, confidence intervals. Do one tiny sample by hand.
  4. Trial runSimulate 100 samples of size 50. What do you see? Does your method run quickly enough for thousands?
  5. Build the theoryWork out the margin as a function of n and p. Say where each step comes from.
  6. Simulate properlyChoose n values, the number of repetitions and a fixed random seed for reproducibility.
  7. Compare formula and simulationGraph both. Where is the gap largest? Why?
  8. Add a twistBias, a different proportion, a finite population or a derivation. Justify your choice.
  9. ReflectWhat does '95% confidence' really claim? Which assumption would worry you most in a real poll?
  10. Write, then cutShow one worked margin calculation and tables for the rest. Define p, p̂ and n once. Check before your teacher's draft.

Maths you'll need

Traps to avoid in this Maths IA

the 95% mistake!
  • Believing '95% confident' means a 95% chance the true value is in your one interval.
  • Using the formula for tiny samples or for p very near 0 or 1 without checking the approximation.
  • Simulating only a few hundred samples and calling the spread exact.
  • Saying a huge sample fixes a biased sample.
  • Quoting the margin formula with no derivation (a weakness at HL).
  • A graph with no label showing whether the margin is a half-width or a full width.
  • Treating a real poll's margin as exact when the polling company's method is unknown.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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