Normal distribution

CIE A-Level Mathematics· Unit 4: Probability & Statistics 1· 15 min read

1. Properties of the Normal Distribution★★☆☆☆⏱ 5 min

📘 Definition

Normal Distribution

$X \sim N (μ, σ^{2})$

A continuous probability distribution with a symmetric, bell-shaped probability density function. $μ$ is the population mean (equal to median and mode due to symmetry), and $σ^{2}$ is the population variance.

Example:

Heights of adult humans, measurement errors, and test scores are commonly modelled as normally distributed.

The normal distribution has several key properties that are frequently tested: it is symmetric about its mean, the total area under the curve equals 1, and almost all (99.7%) of the distribution lies within 3 standard deviations of the mean. The spread of the curve increases as the variance increases.

📐 Worked Example

Identify the mean, variance, and standard deviation of the distribution $X \sim N (30, 36)$ .

1
Recall that the notation $X \sim N (μ, σ^{2})$ lists the mean first, followed by the variance:
2
$\mu = 30, \quad \sigma^2 = 36$
3
Calculate the standard deviation as the square root of the variance:
4
$\sigma = \sqrt{\sigma^2} = \sqrt{36} = 6$
5
Final answer:
6
Mean = 30, Variance = 36, Standard deviation = 6

Exam tip:

Always check if the question asks for variance or standard deviation. Many students lose marks by mixing these up in answers.

2. Standardization and Probability Calculation★★★☆☆⏱ 7 min

📘 Definition

Z-Score

$z = \frac{X - μ}{σ}$

A standardized value that converts any normally distributed variable $X$ to the standard normal variable $Z \sim N (0, 1)$ , which can be looked up in standard normal tables.

Any normal distribution can be converted to the standard normal distribution using the z-score formula. This allows us to use precomputed tables to find cumulative probabilities. The cumulative probability for a z-score $z$ is written $Φ (z) = P (Z \leq z)$ . For negative z-scores, we use the symmetry of the normal distribution: $Φ (- z) = 1 - Φ (z)$ .

📐 Worked Example

Given $X \sim N (50, 100)$ , calculate $P (X < 62)$ .

1
Extract the parameters from the notation:
2
$\mu = 50, \quad \sigma^2 = 100 \implies \sigma = 10$
3
Calculate the z-score for $X = 62$ :
4
$z = \frac{62 - 50}{10} = 1.2$
5
Look up $z = 1.2$ in the standard normal table to get $Φ (1.2) = 0.8849$ , so $P (X < 62) = 0.8849$

📐 Worked Example

For the same distribution $X \sim N (50, 100)$ , calculate $P (X > 45)$ .

1
Calculate the z-score:
2
$z = \frac{45 - 50}{10} = -0.5$
3
Use the symmetry property for negative z:
4
$P(X > 45) = P(Z > -0.5) = 1 - \Phi(-0.5) = \Phi(0.5)$
5
Look up $Φ (0.5) = 0.6915$ , so $P (X > 45) = 0.6915$

Exam tip:

Draw a quick sketch of the normal curve to shade the region you need, this avoids mistakes with tail probabilities.

3. Inverse Normal Distribution Problems★★★★☆⏱ 7 min

Inverse normal problems reverse the process: you are given a probability, and need to find the corresponding value of $X$ . These questions are very common in CIE exams, often worth 5-6 marks.

📘 Definition

Inverse Normal

For a cumulative probability $p = P (X < x)$ , inverse normal finds the value of $x$ that corresponds to the $p$ -th quantile of the normal distribution.

📐 Worked Example

Given $X \sim N (20, 25)$ , find $x$ such that $P (X > x) = 0.10$ .

1
Convert the upper tail probability to a lower tail probability:
2
$P(X < x) = 1 - P(X > x) = 1 - 0.10 = 0.90$
3
Extract parameters:
4
$\mu = 20, \quad \sigma = 5$
5
Find the z-score corresponding to cumulative probability 0.90: $z = 1.282$
6
Rearrange the z-score formula to solve for $x$ :
7
$x = \mu + z\sigma = 20 + (1.282)(5) = 26.41$
8
Final answer (3 significant figures): $x \approx 26.4$

Exam tip:

Always confirm which tail the question is asking for. For example, the upper 10% is the 90th percentile, not the 10th percentile.

4. Common Pitfalls

Wrong move:

Confusing variance and standard deviation when calculating z-scores

Why:

The notation $N (μ, σ^{2})$ gives variance as the second parameter, so students often use variance instead of standard deviation

Correct move:

Always take the square root of the second parameter to get $σ$ before starting any calculations

Wrong move:

Incorrectly calculating probabilities for negative z-scores

Why:

Standard normal tables only give probabilities for positive z, so students often forget to apply the symmetry rule

Correct move:

Always use the identity $Φ (- z) = 1 - Φ (z)$ for negative z-scores

Wrong move:

Using the upper tail probability directly for inverse normal problems

Why:

Students use the given upper tail probability to find z, instead of converting to a lower tail probability

Correct move:

Always convert to a lower cumulative probability $p = P (X < x)$ before looking up z in the table

Wrong move:

Forgetting continuity correction when approximating discrete data

Why:

When using normal distribution to model discrete variables like counts, students forget to adjust the value by 0.5

Correct move:

Always add or subtract 0.5 to the discrete value when applying the normal approximation to a discrete distribution

5. Quick Reference Cheatsheet

Concept	Key Formula / Rule
General Normal Notation	$X \sim N (μ, σ^{2})$
Standard Normal Distribution	$Z \sim N (0, 1)$
Z-score formula	$z = \frac{X - μ}{σ}$
Negative z probability	$Φ (- z) = 1 - Φ (z)$
Inverse normal x	$x = μ + z σ$
Empirical Rule	68% within 1 $σ$ , 95% within 2 $σ$ , 99.7% within 3 $σ$

6. Frequently Asked

Do I need to memorize z-values for common probabilities?

No, CIE provides a standard normal table in the exam, but you should memorize common values like $z = 1.645$ for the 95th percentile to save time.

Is the normal distribution discrete or continuous?

The normal distribution is always a continuous probability distribution, meaning it can take any value within an interval.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 41
Probability calculation question
2023 · 42
Inverse normal problem
2024 · 43
Mixed probability and inverse question

Going deeper

What's Next

The normal distribution is one of the most foundational concepts in statistics, underpinning almost all advanced topics in CIE A-Level Mathematics and beyond. Mastery of the calculations here is required for approximating binomial distributions, constructing confidence intervals, and conducting hypothesis testing in later units. It is also widely used in real-world applications from data science to medical research.

→Hypothesis Testing