IA statistics, step by step · Normal distribution checks

Is a normal model reasonable? Checks for your IA data

AA SLAA HLAI SLAI HL

Many techniques assume normal data. Here are four checks on a sample of 40 apples — a histogram with the fitted normal curve, mean against median, the 68–95–99.7 proportions and expected counts by class.

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

Before a t-test or a confidence interval (both assume normal data).
When you want to use a normal model for probabilities.
When a context suggests natural variation around a typical value (masses, heights, times).

Course fit: Every course: the normal distribution, its properties and normal probability calculations are in the SL core of AA and AI. Testing normality formally with a χ² goodness-of-fit test needs the AI χ² test.

The example data

A student weighed 40 apples of one variety, picked at random from a supermarket's loose display, on kitchen scales reading to the nearest gram.

128, 133, 135, 137, 139, 140, 141, 142, 143, 144, 145, 146, 146, 147, 148, 148, 149, 150, 150, 151, 151, 152, 153, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 168, 170, 173, 178 (g)

Four checks on the apples

The checks use the sample mean and the sample standard deviation s as estimates of the normal model's μ and σ.

Step 1 · Mean, median and standard deviation

n = 40, x̄ = 151.9, median = 151.0, s_n−1 = 11.20

In a normal distribution the mean and median are equal. Here they differ by 0.900, which is 0.080 standard deviations — small, so no strong sign of skew.

Step 2 · The 68–95–99.7 check

Count the values within 1, 2 and 3 standard deviations of the mean and compare with the normal distribution's proportions.

Proportions within k standard deviations
Interval	Count	Percentage of data	Normal model
within 1 s.d.	28	70.0%	68.27%
within 2 s.d.	38	95.0%	95.45%
within 3 s.d.	40	100%	99.73%

Histogram of mass of an apple with a fitted normal curve — The bars are the data; the curve is the normal model with the sample's mean and standard deviation, scaled to the same total.

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

A Criterion A (Presentation): The histogram and the checks next to each other, before the analysis that relies on them.
B Criterion B (Mathematical communication): N(μ, σ²) notation used correctly, with the estimates stated.
C Criterion C (Personal engagement): Questioning your own data's shape before trusting a technique.
D Criterion D (Reflection): Reflecting on how far the normal assumption holds and what follows if it does not.
E Criterion E (Use of mathematics): Correct normal probabilities and expected counts.

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

How do I check if my data are normally distributed for my IA?

Draw a histogram (with the normal curve), compare the mean and median, check the 68–95–99.7 proportions, and if you study AI, consider a χ² goodness-of-fit test on grouped data.

What if my data are not normal?

Say so, and choose techniques that do not need normality (median and IQR, Spearman's rank) or discuss how the departure affects your test. Transforming the data (logs) is another option to explain.

Should I use σ or s for a normal model?

s (dividing by n − 1) is the usual estimate of the population's standard deviation from a sample. State which you used.

Next steps

Related: t-test: two-sample and paired · Chi-squared goodness of fit. Or analyse your own data, find a data set in the IA data bank, and see what the IA package adds.

Free: the IA checklist an examiner uses

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