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:

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.