$X \sim N(\mu, \sigma^2)$ with $P(X > 36.88) = 0.025$ and $P(X < 27.16) = 0.10$. Find $\mu$ and $\sigma$.

1. Convert upper tail: $P(X < 36.88) = 0.975$. 2. Z-scores from Standard Normal: $Z_1 = 1.9599\ldots$ (area 0.975), $Z_2 = -1.2815\ldots$ (area 0.10). 3. Simultaneous equations via $X = \mu + Z\sigma$: $\mu + 1.9599\sigma = 36.88$ and $\mu - 1.2815\sigma = 27.16$. 4. Subtract : $3.2414\sigma = 9.72 \implies \sigma = 2.998\ldots \implies \mathbf{\sigma = 3.00}$. 5. Substitute : $\mu + 1.9599(3) = 36.88 \implies \mathbf{\mu = 31.0}$.

For the standard normal distribution $Z \sim N(0, 1^2)$ , find: $P(Z < 1.5)$ . $P(-2.1 < Z < -0.3)$ . A random variable is given as $X \sim N(2, 0.1^2)$ . By using the standardization formula, re-express the probability $P(X < 2.15)$ in the form $P(Z < a)$ , where $a$ is a constant to be found.

CG50: Ncd with $\mu=0, \sigma=1$ . $\text{Lower} = -1 \times 10^{99}, \text{Upper} = 1.5 \implies 0.93319... \implies 0.933$ . A1 $\text{Lower} = -2.1, \text{Upper} = -0.3 \implies 0.36417... \implies 0.364$ . (M1)A1 Standardize using $Z = \frac{x - \mu}{\sigma}$ . (M1) $Z = \frac{2.15 - 2}{0.1} = \frac{0.15}{0.1} = 1.5$ . Therefore, $P(X < 2.15)$ is exactly $P(Z < 1.5)$ . $a = 1.5$ . A1

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IB Maths AA SL · Unit 4: Statistics and Probability

Normal Distribution (IB Maths AA SL): notes, exam questions and worked examples

Q: Is the normal distribution on Paper 1 or Paper 2?

Calculations need the normal cdf and inverse normal on a GDC, so they belong on Paper 2. Paper 1 can still test the symmetry of the curve, for example that the probability below μ − a equals the probability above μ + a.

Q: Do I need to use z-scores if I have a GDC?

For a probability or a value with μ and σ known, the GDC does it directly. You need z = (x − μ)/σ when the mean or standard deviation is unknown.

Q: How many IB Maths AA SL normal distribution questions are there?

There are 41 exam-style normal distribution questions in the AA SL question bank (Paper 1: 9 · Paper 2: 32), graded 11 easy, 14 medium, 11 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Updated October 2026 · By Pete Bromfield

A normal distribution $X \sim N (μ, σ^{2})$ is a symmetric bell curve centred on the mean $μ$ , with spread set by the standard deviation $σ$ . In AA SL you find probabilities and inverse values on your GDC, and you standardise with $z = \frac{x - μ}{σ}$ when the mean or standard deviation is unknown.

41 questions
Paper 1: 9
Paper 2: 32
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Normal Distribution notes

The normal curve

The curve is symmetric about $μ$ , so $P (X < μ) = 0.5$ , and the total area under it is 1. About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The variance is $σ^{2}$ : the notation $N (50, 10^{2})$ means a standard deviation of 10.

Finding probabilities

Sketch the curve, shade the region you want, then use the normal cdf on your GDC with the lower bound, upper bound, $μ$ and $σ$ . If $X \sim N (50, 10^{2})$ , $P (X > 65) \approx 0.0668$ . For a continuous variable, $P (X < a)$ and $P (X \leq a)$ are the same.

Inverse normal: finding a value

When the probability is given and the value is unknown, use inverse normal. The GDC needs the area to the left: for $P (X > a) = 0.1$ , enter 0.9. With $X \sim N (50, 10^{2})$ and $P (X < a) = 0.9$ , $a \approx 62.8$ .

Standardising: unknown mean or standard deviation

$z = \frac{x - μ}{σ}$ measures how many standard deviations $x$ is from the mean, and $Z \sim N (0, 1)$ . If $X \sim N (μ, 4^{2})$ and $P (X < 30) = 0.8$ , the inverse normal of 0.8 for $Z$ gives $z \approx 0.8416$ , so $\frac{30 - μ}{4} = 0.8416$ and $μ \approx 26.6$ . With both unknown, write two such equations and solve them simultaneously.

Key formulas

Standardised normal In the booklet: $z = \frac{x - μ}{σ}$
Normal: about 68%, 95%, 99.7% of values lie within 1, 2, 3 s.d. of $μ$ Learn it

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Normal distribution

Normal Distribution worked examples

Worked example 1: Using the GDC to find probabilities · easy

$W \sim N (200, {1.75}^{2})$ . Batch of 500 chocolate bars. Expected number weighing over 203 g?

Solution

5 steps · 1 to fill in

Examiner tip:

N (μ, σ^{2})

gives the VARIANCE as the second parameter. Input

σ

(not

σ^{2}

) into your GDC.

Worked example 2: Inverse normal calculations · medium

$P \sim N (11.3, {2.1}^{2})$ kg. The heaviest 7\% are "premium large". Find the minimum weight.

Solution

4 steps · 1 to fill in

Examiner tip: If your GDC only accepts left-tail areas, subtract the upper percentage from 1. Inputting

0.07

directly finds the LIGHTEST 7\%.

Worked example 3: Finding unknown mean and standard deviation · hard

$X \sim N (μ, σ^{2})$ with $P (X > 36.88) = 0.025$ and $P (X < 27.16) = 0.10$ . Find $μ$ and $σ$ .

Solution

5 steps

Examiner tip: Always standardise using

Z

-scores from

N (0, 1)

when reversing to find

μ

σ

. Guess-and-check on the GDC is not valid working.

Play with it The shape of a normal distribution: drag the sliders

The points lie on the curve of a normal distribution. Change m (the mean) and s (the standard deviation) until the curve passes through every point. What does changing m do to the curve? What does changing s do?

Want more? Open the full Grapher.

Normal Distribution exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 4 marks · Paper 1

The distribution of heights of adult women in the UK follows a normal distribution with a mean of $162 cm$ and a standard deviation of $6.3 cm$ . Using the geometric properties of the normal distribution curve (the 68-95-99.7 rule), calculate the approximate range of heights within which:

the central $68 %$ of adult women will fall.
the central $95 %$ of adult women will fall.

Check your answer in practice →

Question 2 · medium · 5 marks · Paper 2

For the standard normal distribution $Z \sim N (0, 1^{2})$ , find:

$P (Z < 1.5)$ .
$P (- 2.1 < Z < - 0.3)$ .
A random variable is given as $X \sim N (2, {0.1}^{2})$ . By using the standardization formula, re-express the probability $P (X < 2.15)$ in the form $P (Z < a)$ , where $a$ is a constant to be found.

Full worked solution Free, no sign-in

CG50: Ncd with $μ = 0, σ = 1$ .
$Lower = - 1 \times 10^{99}, Upper = 1.5 ⟹ 0.93319 . . . ⟹ 0.933$ . A1
$Lower = - 2.1, Upper = - 0.3 ⟹ 0.36417 . . . ⟹ 0.364$ . (M1)A1
Standardize using $Z = \frac{x - μ}{σ}$ . (M1)
$Z = \frac{2.15 - 2}{0.1} = \frac{0.15}{0.1} = 1.5$ .
Therefore, $P (X < 2.15)$ is exactly $P (Z < 1.5)$ . $a = 1.5$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

A machine is used to fill cans of a particular brand of soft drink. The volume, $V$ in ml, of soft drink in the cans is normally distributed with a mean of $330 ml$ and a standard deviation of $σ ml$ .

Given that exactly $15 %$ of the cans contain more than $333.4 ml$ of soft drink, find the value of $σ$ .
Find $P (320 \leq V \leq 340)$ .
Six cans of the soft drink are chosen at random. Find the probability that all six cans contain less than $329 ml$ of soft drink.

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Where students lose marks on normal distribution

Examiner's noteBinomial and normal: "at least", inequalities, rounding bounds and which distribution

Where marks are lost: Recurring errors: P(X ≥ 1) written as 1 − P(X ≤ 1); forgetting to reverse an inequality after dividing by a negative logarithm; an off-by-one slip when turning a strict inequality for a whole-number variable into a cumulative probability; a normal probability found for a single value instead of between the rounding bounds; the binomial used where the normal was needed (or the reverse); an inverse-normal value taken from the wrong tail; and μ and σ from two probabilities solved by hand.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Normal Distribution on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Normal distribution on your calculator

Normal Distribution FAQ

Is the normal distribution on Paper 1 or Paper 2?

Calculations need the normal cdf and inverse normal on a GDC, so they belong on Paper 2. Paper 1 can still test the symmetry of the curve, for example that the probability below μ − a equals the probability above μ + a.

Do I need to use z-scores if I have a GDC?

For a probability or a value with μ and σ known, the GDC does it directly. You need z = (x − μ)/σ when the mean or standard deviation is unknown.

How many IB Maths AA SL normal distribution questions are there?

There are 41 exam-style normal distribution questions in the AA SL question bank (Paper 1: 9 · Paper 2: 32), graded 11 easy, 14 medium, 11 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Next steps

More AA SL Unit 4: Bivariate Data and Correlation · Discrete Random Variables · Measures of Central Tendency · Presentation of Data

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Normal distribution (probability lookups) · Inverse normal (finding cut-off values).

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL statistics and probability knowledge organiser

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Normal Distribution (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Normal Distribution notes

The normal curve

Finding probabilities

Inverse normal: finding a value

Standardising: unknown mean or standard deviation

Key formulas

Normal Distribution worked examples

Worked example 1: Using the GDC to find probabilities · easy

Solution

Worked example 2: Inverse normal calculations · medium

Solution

Worked example 3: Finding unknown mean and standard deviation · hard

Solution

Normal Distribution exam-style questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Where students lose marks on normal distribution

Normal Distribution on your calculator

Normal Distribution FAQ

Next steps

What else is free here