Normal distribution questions on Edexcel IAL S1 follow a small number of patterns, and the mark scheme rewards the same step in every one of them: standardising. This post works through the four patterns with the tables, shows what earns each mark, and ends with the longest of the four: finding an unknown mean and standard deviation.
The one step that carries the method mark
If $X \sim N(\mu, \sigma^2)$, then
$$Z = \frac{X-\mu}{\sigma} \sim N(0,1)$$
and the tables give $\Phi(z) = P(Z<z)$ for $z \geq 0$. The standardising line, with the question's numbers in it, is almost always the first M1. Write it even when your calculator could give the probability directly.
Two facts do the rest:
- $P(Z>z) = 1 - \Phi(z)$
- $\Phi(-z) = 1 - \Phi(z)$, because the curve is symmetric about $0$.
All the examples below use $X \sim N(50, 4^2)$.
Pattern 1: a single inequality
Example 1 · 2 marks
Find $P(X<56)$.
Solution
$P(X<56) = P\left(Z < \dfrac{56-50}{4}\right) = P(Z<1.5)$ (M1)
$= \Phi(1.5) = 0.9332$ (A1)
Example 2 · 2 marks
Find $P(X>45)$.
Solution
$P(X>45) = P\left(Z > \dfrac{45-50}{4}\right) = P(Z>-1.25)$ (M1)
$= P(Z<1.25) = 0.8944$ (A1)
Draw a quick sketch for anything with a negative $z$. The region to the right of $-1.25$ is more than half the curve, so the answer must be above $0.5$. That check catches the slip of giving $1-0.8944$.
Pattern 2: between two values
Example 3 · 3 marks
Find $P(46<X<53)$.
Solution
Standardise both ends: $P(-1 < Z < 0.75)$ (M1)
$= \Phi(0.75) - \Phi(-1) = 0.7734 - (1 - 0.8413)$ (M1)
$= 0.6147$, awrt $0.615$ (A1)
Questions often add a context step: "a machine fills 200 bags; how many would you expect to weigh more than 56 g?" The expected number is $200 \times P(X>56) = 200 \times (1-0.9332) = 13.36$. An expected number does not have to be a whole number, so leave it as $13.4$ unless the question tells you to round.
Pattern 3: inverse problems
Here you are given a probability and asked for a value of $X$. Use the percentage points table, then un-standardise.
Example 4 · 3 marks
Find the value $a$ such that $P(X>a) = 0.05$.
Solution
From the percentage points table, $P(Z>1.6449) = 0.05$. (B1)
$\dfrac{a-50}{4} = 1.6449$ (M1), so $a = 50 + 4 \times 1.6449 = 56.6$ (3 s.f.) (A1)
If the tail is on the left, for example $P(X<b)=0.05$, the $z$-value is $-1.6449$. The table only lists positive values, so write the minus sign yourself; the sketch tells you which side of the mean $b$ must be.
Pattern 4: an unknown mean and standard deviation
This is the longest of the four patterns, and it always leads to a pair of simultaneous equations.
Example 5 · 7 marks
The random variable $Y \sim N(\mu, \sigma^2)$. Given that $P(Y<20) = 0.1$ and $P(Y>40)=0.2$, find $\mu$ and $\sigma$.
Solution
From the percentage points table: $P(Z>1.2816)=0.1$, so $P(Z<-1.2816)=0.1$; and $P(Z>0.8416)=0.2$. (B1 B1)
Standardise each condition:
$$\frac{20-\mu}{\sigma} = -1.2816, \qquad \frac{40-\mu}{\sigma} = 0.8416 \qquad \textbf{(M1 M1)}$$
So $\mu - 1.2816\sigma = 20$ and $\mu + 0.8416\sigma = 40$.
Subtracting: $2.1232\sigma = 20$, so $\sigma = 9.42$ (3 s.f.) (M1 A1)
Then $\mu = 20 + 1.2816 \times 9.4197\ldots = 32.1$ (3 s.f.) (A1)
Three things decide these marks. Use the percentage points table, not a $z$-value read backwards from the $\Phi$ table, because it gives the $z$-values to the four decimal places the working needs. Get the signs right: $20$ is below the mean, so its $z$ is negative. And use the unrounded $\sigma$ when you find $\mu$.
An S1 normal distribution checklist
- Write $Z = \dfrac{X-\mu}{\sigma}$ with the numbers in, every time.
- Sketch the curve and shade the region before you read a table.
- $N(50, 4^2)$ means $\sigma = 4$. If a question writes $N(50, 16)$, $\sigma$ is still $4$.
- Inverse question? Use the percentage points table, then solve for $x$.
- Give probabilities to 4 decimal places from the tables, or 3 significant figures, unless the question says otherwise.
You can practise all four patterns in the IAL S1 normal distribution questions on the S1 unit page, where every question has an Edexcel-style mark scheme, and the courses page links the other IAL units.
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FAQ
Can I use my calculator's normal distribution function in IAL S1?
A calculator is allowed in S1, but the mark scheme gives method marks for standardising, so always write the $z$ calculation, for example $\frac{56-50}{4}=1.5$, before the probability.
What is the difference between the two normal tables?
The first table gives $\Phi(z)=P(Z<z)$ for a value of $z$. The percentage points table goes the other way, giving the $z$ that cuts off a tail probability $p$. Use it for "find the value exceeded by 5%" questions.
What does awrt mean in a mark scheme?
"Answers which round to". An answer of awrt $0.615$ accepts $0.6147$ or $0.615$, but not $0.61$.
Exam-style S1 normal distribution questions, each with an Edexcel-style mark scheme (M, A and B marks).
Practise IAL S1 →