IB Maths AA SL · Unit 4: Statistics and Probability
IB Maths AA SL Normal Distribution Questions
Exam-style IB Maths AA SL normal distribution questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 37 questions
- Paper 1: 9
- Paper 2: 28
- 7 easy
- 13 medium
- 10 hard
- 3 very hard
- 4 starter
- 3 worked examples
Practise Normal Distribution questions →
AA SL formula booklet
What you need to know
Standardising with z = (x - μ)/σ, then reading off the standard normal. SL AA Paper 2 always asks a working-backwards question — given P(X < a) = 0.95, find a. Normal distribution and z-scores overview →
What's examined in AA SL normal distribution
The question bank covers these normal distribution question types (number of questions in brackets):
- Finding Unknown Parameters (18)
- Calculating Probabilities (13)
- Properties and Z-scores (6)
Key formulas
- Standardising a normal variable
- \(Z = \dfrac{X - \mu}{\sigma}\)
In the same notation as the IB formula booklet. All AA SL formulas →
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?
1. Parameters: $\mu = 200$, $\sigma = 1.75$.
2. Probability required: $P(W > 203)$.
3. GDC Normal CD: Lower $= 203$, Upper $= 10^{99}$, $\sigma = 1.75$, $\mu = 200$.
4. Result: $P(W > 203) = 0.04323\ldots$
5. Expected bars: $500 \times 0.04323 = 21.61\ldots \implies \mathbf{22}$ bars.
Examiner tip: $N(\mu, \sigma^2)$ gives the VARIANCE as the second parameter. Input $\sigma$ (not $\sigma^2$) 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.
1. Left-tail area: $1 - 0.07 = 0.93$.
2. Set up: find $k$ with $P(P < k) = 0.93$.
3. GDC InvN: Area $= 0.93$, $\sigma = 2.1$, $\mu = 11.3$.
4. Result: $k = 14.398\ldots \implies \mathbf{14.4}$ kg.
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
$X \sim N(\mu, \sigma^2)$ 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: $\mu + 1.9599(3) = 36.88 \implies \mathbf{\mu = 31.0}$.
Examiner tip: Always standardise using $Z$-scores from $N(0, 1)$ when reversing to find $\mu$ or $\sigma$. Guess-and-check on the GDC is not valid working.
Try these IB Maths AA SL normal distribution questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are 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 \(162\text{ cm}\) and a standard deviation of \(6.3\text{ cm}\).
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.
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.
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 \(330\text{ ml}\) and a standard deviation of \(\sigma\text{ ml}\).
Given that exactly \(15\%\) of the cans contain more than \(333.4\text{ ml}\) of soft drink, find the value of \(\sigma\).
Find \(P(320 \le V \le 340)\).
Six cans of the soft drink are chosen at random. Find the probability that all six cans contain less than \(329\text{ ml}\) of soft drink.
All 37 normal distribution questions with mark schemes →
FAQ
How many IB Maths AA SL normal distribution questions are there?
There are 37 exam-style normal distribution questions in the AA SL question bank (Paper 1: 9 · Paper 2: 28), graded 7 easy, 13 medium, 10 hard, 3 very hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is normal distribution on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 9 · Paper 2: 28. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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