# The Normal Distribution

Mathematics I · Data Analysis and Probability · https://tryals.app/learn/mathematics-i/the-normal-distribution

## The Bell Curve

The **normal distribution** (Gaussian) is the most important continuous distribution in statistics. Its symmetric, bell-shaped density is written $N(\mu, \sigma^2)$ and is fully determined by two parameters:

| Parameter | Role | Effect |
| --- | --- | --- |
| **Mean** $\mu$ | Centre | Where the peak sits |
| **Standard deviation** $\sigma$ | Spread | Larger $\sigma$ means a wider, flatter bell |

It is ubiquitous due to the Central Limit Theorem: sums of independent effects tend toward normality. Measurement errors and heights are approximately normal.

Its shape gives the **empirical rule (68–95–99.7)** for how data cluster:

- About **68%** lie within **1** standard deviation ($\mu \pm \sigma$).
- About **95%** lie within **2** standard deviations ($\mu \pm 2\sigma$).
- About **99.7%** lie within **3** standard deviations ($\mu \pm 3\sigma$).

Beyond $2\sigma$ is rare; beyond $3\sigma$ is very rare.

## Standardisation

**Standardisation** converts any normal value to a **z-score** using the formula:

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

Here, $z$ is the number of standard deviations $x$ lies from the mean. This maps any $N(\mu, \sigma^2)$ onto the **standard normal** $N(0, 1)$, letting a single table handle all probabilities.

A **z-score** of $+2$ means two standard deviations above average, giving a universal, unit-free way to compare values across different distributions.

> **Common pitfall:** Confusing the **z-score** with a raw value or a probability. A z-score is the number of **standard deviations from the mean**, not the value itself and not a probability. To get a probability, you must convert the z-score through the standard normal.

## Practice questions

3 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. Heights, measurement errors and exam scores are all roughly normal. What do these have in common that produces the same curve?

A. They are all measured on continuous scales, which produces normality
B. Each is a sum of many small independent influences
C. They are all bounded below by zero, which forces a symmetric shape
D. They were each selected as examples because they happen to be normal

**Answer:** B. Each is a sum of many small independent influences

**Why:** Height is many genes plus nutrition plus health; an exam score aggregates many items. The theorem's indifference to the underlying distributions is what makes the curve ubiquitous, and knowing the condition tells you when to expect something else: multiplicative influences give lognormal, which is why income is skewed.

Page: https://tryals.app/practice/mathematics-i/the-normal-distribution/heights-measurement-errors-and-exam-scores-are-all-roughly-normal

### 2. Select every true statement about the normal distribution.

A. Its mean, median and mode all coincide
B. A larger standard deviation raises the height of the peak
C. It is completely determined by its mean and standard deviation
D. About 95% of values lie within 1 standard deviation of the mean
E. It is symmetric about its mean

**Answer:** A. Its mean, median and mode all coincide; C. It is completely determined by its mean and standard deviation; E. It is symmetric about its mean

**Why:** 68% lies within one standard deviation and 95% within two, so option 4 swaps them. A larger standard deviation _lowers_ the peak, since the fixed total area must spread further. The first three are exactly the defining properties.

Page: https://tryals.app/practice/mathematics-i/the-normal-distribution/select-every-true-statement-about-the-normal-distribution

### 3. The Central Limit Theorem explains why sums of many effects are normal, which is why income and city sizes are normal too.

**Answer:** False

**Why:** False. The CLT applies to *sums* of comparable independent effects. Income and city size grow multiplicatively, proportional increases compounding, which produces a log-normal distribution with a long right tail, not a normal one. The mechanism decides the distribution, not the number of factors.

Page: https://tryals.app/practice/mathematics-i/the-normal-distribution/the-central-limit-theorem-explains-why-sums-of-many-effects-are