# Normal distribution

> IB Mathematics: Analysis and Approaches HL · Statistics & Probability
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u4-normal-distribution/

The normal distribution is the most widely used continuous probability distribution, describing many natural and man-made phenomena. This guide covers its key properties, probability calculations, and inverse normal applications for IB AA HL.

**Prerequisites:** [Continuous probability distributions](https://www.owlsprep.com/study/ib-math-aa-hl-u4-continuous-probability/); Basic probability rules

## Learning objectives

- Recall key properties of the normal distribution curve
- Standardize normal variables and calculate cumulative probabilities
- Use inverse normal to find quantiles for given probabilities
- Solve for unknown mean or standard deviation in HL problems

## Properties of the Normal Distribution

**Normal Distribution** — A symmetric, bell-shaped continuous probability distribution, parameterized by its mean $\mu$ (center) and variance $\sigma^2$ (spread).

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

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

The normal probability density function is symmetric about $\mu$, 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.

> **Empirical Rule Mnemonic**
>
> 68 with one, 95 with two, 99.7 with three. All values are within 1, 2, 3 standard deviations from the mean.

**Worked example:** For $X \sim N(75, 10^2)$, what percentage of observations lie between 55 and 95?

1. Identify parameters: $\mu = 75$, $\sigma = 10$.
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.

## Calculating Probabilities via Standardization

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

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

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: $\mu = 30$, $\sigma = 4$ (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 $P(X < 36) \approx 0.933$.

**Check your understanding**

Check your understanding

1. What is the Z-score for $x=22$ in the example above?

   - A) 2
   - B) -2
   - C) -1.5
   - D) 1.5

   *Why:* Correct: $z = (22 - 30)/4 = -2$

## Inverse Normal Calculations

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.

**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 $\mu$ and $\sigma$.

**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.

> **tip**
>
> Always confirm if you need the lower or upper tail. For top/upper p%, always use cumulative probability 1-p.

## Solving for Unknown $\mu$ or $\sigma$ (HL)

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 $X \sim N(\mu, 12^2)$, 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 $\mu$:
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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using variance $\sigma^2$ instead of standard deviation $\sigma$ in the Z-score formula.
  - Why it fails: Normal distribution notation uses variance as the second parameter, so it is common to forget to take the square root.
  - Correct: Always confirm you are using standard deviation, the square root of the given variance, when calculating Z-scores.
- **Wrong:** Using p as the cumulative probability for upper tail inverse normal problems.
  - Why it fails: Most calculators and tables return lower tail cumulative probabilities by default.
  - Correct: For the top p% of values, use cumulative probability $1-p$ when calculating the Z-score or inverse normal result.
- **Wrong:** Calculating $P(X = k)$ for a normal distribution.
  - Why it fails: The normal distribution is continuous, so the probability of any single exact value is 0.
  - Correct: 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:** Assuming any symmetric bell-shaped data is normally distributed.
  - Why it fails: Many distributions are symmetric but not normal, so you cannot assume normality unless it is stated.
  - Correct: Only use normal distribution methods when the question explicitly says the variable is normally distributed.
- **Wrong:** Forgetting the negative sign on Z-scores for values below the mean.
  - Why it fails: Tables often only show positive Z-scores, leading to missing negative signs.
  - Correct: Double-check your Z-score sign: values below the mean have negative Z-scores, values above have positive.

## Cheatsheet

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

## What's next

The normal distribution is a foundational concept in probability and is frequently tested in IB AA HL, often combined with other topics such as conditional probability. Note that confidence intervals, hypothesis testing, the central limit theorem and the normal approximation to the binomial belong to Math AI, not AA, and are not required for AA HL. 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](https://www.owlsprep.com/study/ib-math-aa-hl-u4-bivariate-data-correlation-and-regression/)

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