Updated · By Pete Bromfield, IB examiner

IA statistics, step by step · Confidence intervals

A confidence interval for a mean, step by step

AI HL

How much water does a typical student drink in a school day? A sample mean gives one number; a confidence interval says how precise it is. Worked by hand here, with what the interval does and does not mean.

Guidance based on the checklist Pete Bromfield — an IB examiner and former Head of Maths — uses with his own students. Not affiliated with or endorsed by the IB.

Example data, invented for this guide. The context is realistic, but the numbers were made up to show the method. Use your own collected or sourced data in your IA.

When to use it

  • You want to estimate a population mean from a random sample.
  • The data are roughly normal, or the sample is large.
  • You want to compare an estimate with a published figure or a recommendation.

Course fit: AI HL: confidence intervals for the mean of a normal population are in the AI HL content. They are not in the SL courses or AA — SL and AA students can use them if they explain them.

The example data

Fifteen students, chosen at random from a year group, measured how much water they drank on one school day with a marked bottle.

1.2, 1.8, 2.1, 1.5, 2.4, 1.9, 1.6, 2, 2.2, 1.4, 1.7, 2.5, 1.8, 2, 1.6 (L)

A 95% confidence interval for the mean

The population standard deviation is unknown, so the interval uses s and the t-distribution.

Step 1 · The sample

n = 15,   x̄ = 1.8467 L,   sn−1 = 0.3642 L

The population standard deviation is not known, so the interval uses the t-distribution with ν = n − 1 = 14 degrees of freedom.

Step 2 · Standard error and critical value

Standard error = s / √n = 0.09404

t* for 95% confidence = 2.145

Box plot of water drunk in a day for the example sample
Look at the shape before trusting a t interval: roughly symmetric, no strong outliers.

In the full worked analysis

  • The rest of the working: steps 3 to 3
  • What the example shows, in context
  • On a GDC: TI-84 Plus CE, TI-Nspire CX and Casio fx-CG50
  • What examiners look for
  • Common mistakes
  • Limitations to discuss

How this maps to the IA criteria

These are the current criteria A–E, for exams up to November 2028. For the new courses (first assessment May 2029) the IB has confirmed one set of four criteria for SL and HL: A Problem specification (4 marks), B Abstraction (6), C Computation (4) and D Interpretation (6), still 20 marks and 20% of the grade at both levels — see the IB's new AA and AI subject briefs. The detailed descriptors come with the new guide; check with your teacher which criteria apply to you. The advice is our summary, not the IB's wording.

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Frequently asked questions

Are confidence intervals in IB Maths?

Confidence intervals for a population mean are in AI HL. Other courses may use them in an IA if they explain them.

How do I interpret a 95% confidence interval?

In repeated sampling, 95% of intervals made this way contain the true mean. Avoid saying there is a 95% chance this interval contains it.

Should I use z or t for my confidence interval?

Use t when the population standard deviation is unknown and estimated by s from the sample — the usual case in an IA.

Next steps

Related: Is my data normal? · t-test: two-sample and paired. Or analyse your own data, find a data set in the IA data bank, and see what the IA package adds.

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