# The Normal Distribution

> Edexcel International A-Level Mathematics · Edexcel IAL Maths S1 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s1-the-normal-distribution/

This guide covers all Edexcel IAL S1 content for the normal distribution, including probability calculations using standardisation, use of provided statistical tables, and solving for unknown mean and standard deviation via simultaneous equations.

**Prerequisites:** [Ability to rearrange algebraic equations and solve simultaneous linear equations](https://www.owlsprep.com/study/edexcel-gcse-maths-algebra-simultaneous-equations/); [Understanding of basic probability and cumulative distribution functions](https://www.owlsprep.com/study/edexcel-ial-maths-s1-probability-distributions/)

## Learning objectives

- Recall the symmetry, mean and variance properties of the normal distribution
- Standardise normal variables using $Z=(X-μ)/σ$ and use provided statistical tables to calculate probabilities
- Set up and solve simultaneous equations to find unknown $μ$ and/or $σ$ from two probability statements

## Key Properties of the Normal Distribution

**Normal Distribution** — A continuous probability distribution with a symmetric, bell-shaped curve, centred at its mean $μ$, with spread determined by variance $σ^2$.

*Notation:* X \sim N(\mu, \sigma^2)

*Example:* Heights of adult women are commonly modelled as $N(165, 7^2)$ cm.

The normal curve is fully symmetric about $μ$, so $P(X < μ) = P(X > μ) = 0.5$, and $P(X < μ - a) = P(X > μ + a)$ for any positive $a$. You do not need to know the formula for the probability density function for S1.

> **Exam Context**
>
> All normal distribution questions in S1 allow calculators (CAS not permitted) and assume you use the provided Edexcel Mathematical Formulae & Statistical Tables booklet.

**Worked example:** State two key properties of the normal distribution curve.

1. The curve is symmetric about the mean $μ$, with half the probability on either side of the mean.
2. The total area under the curve equals 1, representing the total probability of all possible outcomes.

*Calculator:* allowed

## Standardisation & Probability Calculations

**Standard Normal Distribution** — The normal distribution with mean 0 and variance 1, used to standardise any normal variable $X \sim N(μ, σ^2)$ using the formula $Z = (X - μ)/σ$.

*Notation:* Z \sim N(0, 1)

The cumulative distribution function of $Z$ is written $Φ(z) = P(Z \leq z)$, values of which are tabulated in your formula booklet. For any given normal variable, convert to $Z$ first to look up probabilities.

**Worked example:** Given $X \sim N(50, 4^2)$, calculate $P(X \leq 54.2)$. Give your answer to 4 decimal places.

1. First, standardise the value $X=54.2$ using the standardisation formula:
2. $$Z = \frac{54.2 - 50}{4} = 1.05$$
3. Look up $Φ(1.05)$ in the standard normal table: this gives 0.8531.
4. Thus, $P(X \leq 54.2) = 0.8531$, rounded to 4 decimal places as required.

> **tip**
>
> For $P(X > a)$, use $1 - Φ((a - μ)/σ)$. For $P(a < X < b)$, calculate $Φ((b - μ)/σ) - Φ((a - μ)/σ)$.

**Worked example:** Given $X \sim N(80, 3^2)$, find the value of $k$ such that $P(X > k) = 0.05$. Give your answer to 3 significant figures.

1. Rewrite the probability in terms of $Z$: $P(Z > (k - 80)/3) = 0.05$
2. Look up the $z$ value for a 5% upper tail in the percentage points table: this is 1.6449.
3. $$\frac{k - 80}{3} = 1.6449$$
4. $$k = 80 + 3 \times 1.6449 = 84.9347 \approx 84.9$$

*Calculator:* allowed

## Finding Unknown $μ$ and $σ$

Exam questions often give you two probability statements about a normal variable with unknown $μ$ and/or $σ$. You will need to convert these to standard normal form, then solve the resulting simultaneous equations to find the unknown parameters.

**Worked example:** A random variable $X$ is normally distributed. Given $P(X < 72) = 0.8643$ and $P(X < 57) = 0.1151$, find the values of $μ$ and $σ$. Give your answers to 3 significant figures.

1. First, find the corresponding $z$ values from the standard normal table: $Φ(1.1) = 0.8643$ and $Φ(-1.2) = 0.1151$.
2. $$\frac{72 - \mu}{\sigma} = 1.1 \quad (1)$$
3. $$\frac{57 - \mu}{\sigma} = -1.2 \quad (2)$$
4. Subtract equation (2) from equation (1) to eliminate $μ$:
5. $$\frac{15}{\sigma} = 2.3 \implies \sigma = \frac{15}{2.3} \approx 6.52$$
6. Substitute $σ \approx 6.52$ back into equation (1) to solve for $μ$:
7. $$72 - \mu = 1.1 \times 6.52 \approx 7.17 \implies \mu \approx 64.8$$
8. Check your answers by substituting back into the original probability statements to confirm they match the given values.

> **Exam tip**
>
> Always show every step of your simultaneous equation working, as method marks are awarded even if your final answer is incorrect.

*Calculator:* allowed

## Full Exam Style Question Walkthrough

**Worked example:** The times taken for students to complete a maths test are normally distributed with mean $μ$ minutes and standard deviation 8 minutes. Given that 20% of students finish in less than 40 minutes, find (a) the value of $μ$, (b) the percentage of students who take more than 60 minutes to finish. Give all answers to 3 significant figures.

1. Let $T = $ time taken to complete the test, so $T \sim N(μ, 8^2)$.
2. Part (a): We know $P(T < 40) = 0.2$. From the percentage points table, the $z$ value for a lower tail of 0.2 is -0.8416.
3. $$\frac{40 - \mu}{8} = -0.8416 \implies 40 - \mu = -6.7328 \implies \mu \approx 46.7$$
4. Part (b): Calculate $P(T > 60)$ using $μ = 46.7$:
5. $$Z = \frac{60 - 46.7}{8} \approx 1.6625$$
6. Look up $Φ(1.66) = 0.9515$ from the standard normal table, so $P(Z > 1.6625) \approx 1 - 0.9515 = 0.0485$.
7. Convert to percentage: $0.0485 \times 100 = 4.85% \approx 4.9%$ of students take more than 60 minutes.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using variance $σ^2$ instead of standard deviation $σ$ in the standardisation formula
  - Why it fails: The normal distribution notation $X \sim N(μ, σ^2)$ uses variance as the second parameter, so students often mix up variance and standard deviation.
  - Correct: Always take the square root of the given variance value before using it in the $Z=(X-μ)/σ$ formula.
- **Wrong:** Interpolating values from the $Φ(z)$ table for S1 questions
  - Why it fails: Edexcel explicitly states interpolation is not required for S1, and using the closest table value is accepted for full marks.
  - Correct: Use the nearest tabulated $Φ(z)$ value, or the percentage points table for inverse normal calculations.
- **Wrong:** Forgetting symmetry when working with lower tail probabilities < 0.5
  - Why it fails: The standard normal table only lists positive $z$ values, so negative $z$ values require using the property $Φ(-z) = 1 - Φ(z)$.
  - Correct: Use negative $z$ values for probabilities less than 0.5, and apply the symmetry rule to look up values from the table.
- **Wrong:** Using the wrong sign for $z$ when setting up equations for unknown $μ$ and $σ$
  - Why it fails: If an $X$ value is below the mean, the corresponding $z$ value is negative, which students often mix up with positive values.
  - Correct: Check if the given $X$ value is above or below the expected mean, and ensure the sign of your $z$ value matches this relationship.
- **Wrong:** Rounding intermediate calculation values too early
  - Why it fails: Early rounding leads to inaccuracies in final answers, which can cost accuracy marks in exams.
  - Correct: Keep at least 4 decimal places for all intermediate $z$ values and calculations, only round final answers to 3 significant figures or 4 decimal places as required.

## Cheatsheet

| Task | Formula / Method | Exam Notes |
| --- | --- | --- |
| Calculate $P(X \leq a)$ for $X \sim N(μ, σ^2)$ | $Z = (a - μ)/σ$, find $Φ(Z)$ from tables | Round final probability to 4 decimal places |
| Calculate $P(X > a)$ | $1 - Φ((a - μ)/σ)$ | Use symmetry for left/right tails |
| Find $a$ given $P(X \leq a) = p$ | Find $z$ from percentage points table, $a = μ + zσ$ | Use negative $z$ for $p < 0.5$ |
| Find unknown $μ$ and $σ$ | Set up 2 standard normal equations, solve simultaneously | Substitute answers back to check |
| Standard notation | $X \sim N(μ, σ^2)$ | Second parameter is variance, not standard deviation |

## What's next

Now that you have mastered the normal distribution for Edexcel IAL S1, you are ready to practice with full past paper questions to reinforce your skills. Remember that normal approximation to binomial/Poisson distributions and linear combinations of normal variables are not part of the S1 syllabus, and will be covered in later units S2 and S3 respectively. Ensure you are confident using the provided Edexcel statistical tables without interpolation, as this is the explicit expectation for your IAS exam. Always show all working for standardisation and simultaneous equation steps to secure full method marks.

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