Normal distribution

IB Mathematics: Analysis and Approaches HL· 45 min read

1. Properties of the Normal Distribution★★☆☆☆⏱ 15 min

📘 Definition

Normal Distribution

$X \sim N (μ, σ^{2})$

A symmetric, bell-shaped continuous probability distribution, parameterized by its mean $μ$ (center) and variance $σ^{2}$ (spread).

Example:

Heights of adult humans, measurement errors, and test scores all typically follow a normal distribution.

The normal probability density function is symmetric about $μ$ , with the highest point at the mean, and total area under the curve equal to 1. The empirical rule describes the proportion of data within 1, 2, and 3 standard deviations of the mean.

📐 Worked Example

For $X \sim N (75, 1 0^{2})$ , what percentage of observations lie between 55 and 95?

1
Identify parameters: $μ = 75$ , $σ = 10$ .
2
Rewrite bounds in terms of mean and standard deviation:
3
$55 = 75 - 2(10), \quad 95 = 75 + 2(10)$
4
By the empirical rule, 95% of data lies within 2 standard deviations of the mean, so 95% of observations are between 55 and 95.

2. Calculating Probabilities via Standardization★★★☆☆⏱ 20 min

📘 Definition

Standard Normal Distribution

$Z \sim N (0, 1)$

A special case of the normal distribution with mean 0 and variance 1, used to standardize any normal variable to compare across different scales.

To calculate probabilities for any normal distribution, we convert our observed value $x$ to a Z-score using the standardization formula:

z = \frac{x - \mu}{\sigma}

For continuous distributions like the normal, $P (X < k) = P (X \leq k)$ , since the probability of a single exact value is 0.

📐 Worked Example

Given $X \sim N (30, 4^{2})$ , find $P (X < 36)$ .

1
Identify parameters: $μ = 30$ , $σ = 4$ (take square root of variance 16).
2
Calculate the Z-score:
3
$z = \frac{36 - 30}{4} = 1.5$
4
Find the cumulative probability for $Z < 1.5$ from tables or a calculator: $P (Z < 1.5) \approx 0.933$ . So $P (X < 36) \approx 0.933$ .

✓ Quick check

Check your understanding

What is the Z-score for $x = 22$ in the example above?
- A) 2
- B) -2
- C) -1.5
- D) 1.5
Reveal answer
B —
Correct: $z = (22 - 30) /4 = - 2$

3. Inverse Normal Calculations★★★☆☆⏱ 20 min

Inverse normal is the reverse process of probability calculation: you start with a cumulative probability and find the corresponding $x$ -value. This is one of the most common exam question types for normal distribution.

📘 Definition

Inverse Normal Distribution

A function that returns the value of $x$ for a given cumulative probability $P (X < x) = p$ , for a normal distribution with specified $μ$ and $σ$ .

📐 Worked Example

The volume of soda in cans is normally distributed with mean 330ml and standard deviation 5ml. Find the minimum volume of the top 15% of cans.

1
Define the distribution: $X \sim N (330, 5^{2})$ . We need $x$ such that $P (X > x) = 0.15$ , so $P (X < x) = 1 - 0.15 = 0.85$ .
2
Find the Z-score for cumulative probability 0.85: $z \approx 1.036$ .
3
Rearrange the Z-score formula to solve for $x$ :
4
$x = \mu + z\sigma = 330 + (1.036)(5) \approx 335.2$
5
The minimum volume of the top 15% of cans is approximately 335ml.

4. Solving for Unknown $\mu$ or $\sigma$ (HL)★★★★☆HL only⏱ 25 min

✓ Calculator OK

An advanced HL-exclusive question type asks you to find an unknown mean or standard deviation when given a probability and corresponding value. This requires setting up an equation using the Z-score formula and solving for the unknown parameter.

📐 Worked Example

The time taken to complete an exam is normally distributed with standard deviation 12 minutes. 10% of students finish in less than 80 minutes. Find the mean time.

1
We know $X \sim N (μ, 1 2^{2})$ , and $P (X < 80) = 0.10$ .
2
Find Z-score for cumulative probability 0.10: $z \approx - 1.282$ .
3
Substitute into Z-score formula:
4
$-1.282 = \frac{80 - \mu}{12}$
5
Solve for $μ$ :
6
$80 - \mu = -15.384 \implies \mu = 80 + 15.384 \approx 95.4$
7
The mean time to complete the exam is approximately 95.4 minutes.

5. Common Pitfalls

Wrong move:

Using variance $σ^{2}$ instead of standard deviation $σ$ in the Z-score formula.

Why:

Normal distribution notation uses variance as the second parameter, so it is common to forget to take the square root.

Correct move:

Always confirm you are using standard deviation, the square root of the given variance, when calculating Z-scores.

Wrong move:

Using p as the cumulative probability for upper tail inverse normal problems.

Why:

Most calculators and tables return lower tail cumulative probabilities by default.

Correct move:

For the top p% of values, use cumulative probability $1 - p$ when calculating the Z-score or inverse normal result.

Wrong move:

Calculating $P (X = k)$ for a normal distribution.

Why:

The normal distribution is continuous, so the probability of any single exact value is 0.

Correct move:

If you need the probability of a rounded discrete value (e.g. 180cm), use $P (179.5 < X < 180.5)$ for the continuous approximation.

Wrong move:

Assuming any symmetric bell-shaped data is normally distributed.

Why:

Many distributions are symmetric but not normal, so you cannot assume normality unless it is stated.

Correct move:

Only use normal distribution methods when the question explicitly says the variable is normally distributed.

Wrong move:

Forgetting the negative sign on Z-scores for values below the mean.

Why:

Tables often only show positive Z-scores, leading to missing negative signs.

Correct move:

Double-check your Z-score sign: values below the mean have negative Z-scores, values above have positive.

6. Quick Reference Cheatsheet

Concept	Key Formula/Rule
Normal distribution notation	$X \sim N (μ, σ^{2})$
Z-score transformation	$z = \frac{x - μ}{σ}$
Empirical rule	68% (1 $σ$ ), 95% (2 $σ$ ), 99.7% (3 $σ$ )
Inverse normal for top p%	Use cumulative probability $1 - p$
Standard normal	$Z \sim N (0, 1)$
Probability for continuous	$P (X < k) = P (X \leq k)$

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.

2023 · Paper 1
Inverse normal probability calculation
2022 · Paper 2
Solve for unknown mean and standard deviation
2021 · Paper 1
Probability between two values

What's Next

The normal distribution is a foundational concept that underpins almost all inferential statistics, from confidence intervals to hypothesis testing. In IB AA HL, it is frequently combined with other topics including binomial distribution (for normal approximation of binomial variables, an HL topic) and the central limit theorem for sampling distributions. Mastery of normal distribution calculations is critical for both Paper 1 and Paper 2, and forms the base of any future study of statistics at university.

→Bivariate data: correlation and regression