Maths IA · Statistics and probability · topic 18 of 40

Sleep and test scores: how strong is the evidence? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle, but ethics come FIRST

Everyone says "get more sleep". You're going to ask how strong the link to marks really is, and how sure you can be that it isn't just chance.

Difficulty●○○ Gentle
KitAnonymous survey, spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAI, SL and HL
  • ✓ you want your own survey data
  • ✓ you like scatter graphs
  • ✓ you can handle consent and anonymity

The big idea

You collect (with consent, anonymously) hours of sleep and a test score for each student, then measure how strongly they are linked. Correlation gives a number; regression gives a line; a hypothesis test or confidence interval says how much to trust it. The best explorations ask harder questions: is a straight line even sensible, what do outliers do, and does a link show cause or just association? At HL, go beyond reading r off a screen: justify the test's conditions, compare models, or derive the least-squares line.

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.

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Test whether the link is real

Carry out a hypothesis test on the correlation coefficient (AI HL content) and discuss its conditions.

→ Your conclusion has a p-value, but you must check conditions such as roughly bivariate normal data and independence.
Compare a line and a curve

Fit a straight line and a model that levels off, and compare residuals.

→ You discuss which model is more sensible, not which has the higher r², and explain why too many hours might not help.
Derive the line yourself

Show how the least-squares line is found from minimising the sum of squared residuals.

→ This adds the justification HL criterion E rewards; a calculator-only fit does not.
Include a third variable

Record study time or screen time too and see whether the sleep link changes.

→ You meet confounding and have to decide how far a two-variable model can go.
Rank correlation

Use Spearman's rank correlation as well as Pearson.

→ You compare two measures of association and discuss when each is appropriate, especially with outliers.
Quantify the uncertainty

Give a confidence interval for the gradient or for r, and bootstrap it. Both are beyond the course, so introduce them.

→ The conclusion moves from 'there is a link' to 'the link is between these values'.

Your Maths IA research question: run the check

association, not cause!

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

  • Names the two variables and how each is measured (self-reported hours? which test, and out of how many marks?).
  • Says who the sample is and how many, with a plan for consent and anonymity.
  • Says what 'how strong' will mean: a coefficient, a gradient, an interval.
  • Does not claim cause; asks about association.
  • Is narrow enough to carry out in a few weeks with enough responses (30 at the very least).

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The picture to draw

A scatter graph with one dot per student, a line of best fit, and the vertical gap (residual) that the least-squares method makes as small as possible.

hours of sleeptest score (%)each dot = one student, anonymousline of best fit: what doesthe gradient mean?residual: plot these toooutlier: keep it or drop it?say why

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 this question matters to you and your school, and what you expect the pattern to be.
Ethics and data collection~1.5 pages
Consent, anonymity, how sleep and score were measured, the sample, and what you did with missing answers.
Summary of the data~2 pages
Means, spread, a scatter graph with titled axes, and a first look at any outliers.
Correlation~2 pages
Definition of r, your value, what it means in words, and its limits.
Regression~2.5 pages
The model, how the line is found, the gradient and intercept in context, and what the line cannot tell you.
Testing the evidence~2 pages
Hypotheses, conditions, p-value or interval, and an honest statement of what the test does and does not show.
Checking the model~2 pages
Residual plot, outliers, another model, and a judgement on which is better.
Conclusion~1 page
How strong the evidence is, in numbers and in words.
Reflection and limits~1.5 pages
Self-reported sleep, a single test, confounders, sample size, and a next step with a reason.
Bibliographynot counted
Sources for the methods and any sleep research you cite.

What your Maths IA graph could look like

two models = a judgement to make

Made-up survey data, to show the form of a strong graph. Yours will look different. example only

45678910405060708090100Sleep per night / hours (± 0.5 h, self-reported)Test score / %least-squares linelevelling-off modelstudents (made up)two models agree in the middlebut disagree at 9 hours.Which would you trust there?ILLUSTRATIVE · MADE-UP DATA
  • Axes show units and the measurement uncertainty, here the self-reporting.
  • No error bars on the score, because it is a single result; but the horizontal bars remind the reader that sleep is estimated.
  • Two models on one graph invite a comparison and a judgement.
  • The scale starts at 40 and 4, not zero; say so in the text.
  • Another graph to consider: residuals against hours slept, which should look like random scatter if a line is appropriate.

Your game plan

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

0 of 10 done

  1. Plan the ethics firstDecide how you will get consent, keep responses anonymous and store them. Ask your teacher before you collect anything.
  2. Choose the measuresHours of sleep (usual night? last night?) and a single comparable score. Say how each is defined.
  3. Learn the toolsPearson r, least-squares regression, what a p-value or confidence interval says. Do one tiny example by hand.
  4. PilotTest your survey on 5 classmates. Are the questions clear? Is the score comparable between students?
  5. Collect and cleanGather at least 30 responses. Decide in advance how you will handle missing or impossible values.
  6. ExploreScatter graph and summary statistics first. What do you see? Write a prediction for r before you calculate it.
  7. Fit and testCalculate r and the line. Check the conditions for any test. Interpret the gradient in context.
  8. Check the modelResiduals, outliers and a second model. Reflect on any change in your thinking.
  9. Reflect and concludeSay how strong the evidence is and where the doubt remains. Point to causes beyond sleep.
  10. Write, then cutShow one worked calculation, then tables. Check every variable is defined, then get your teacher's draft read.

Maths you'll need

Traps to avoid in this Maths IA

a classic: r² alone
  • Collecting names or identifiable data, or skipping consent.
  • Claiming that more sleep causes higher marks.
  • Quoting r with no check that a straight line suits the data.
  • Dropping an outlier without a reason or a before-and-after comparison.
  • A tiny sample (under 30) and large claims.
  • Judging a model by r² alone.
  • At HL: calculator regression only; criterion E expects justification, a derivation, or a test with its conditions checked.

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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