The Normal Distribution

The Bell Curve

The normal distribution (Gaussian) is the most important continuous distribution in all of statistics. Its density is the symmetric, bell-shaped curve, fully determined by two parameters: the mean $\mu$ (its centre, where the peak sits) and the standard deviation $\sigma$ (its spread — larger $\sigma$ means a wider, flatter bell). It is written $N(\mu, \sigma^2)$.

The normal distribution is ubiquitous because of the Central Limit Theorem (next lesson): sums and averages of many independent effects tend toward normality, so measurement errors, heights, and countless natural quantities are approximately normal.

Its symmetric shape gives the memorable empirical rule (68–95–99.7) for how data cluster around the mean:

• About 68% of values lie within 1 standard deviation of the mean ($\mu \pm \sigma$).
• About 95% lie within 2 standard deviations ($\mu \pm 2\sigma$).
• About 99.7% lie within 3 standard deviations ($\mu \pm 3\sigma$).

This lets you judge how unusual a value is at a glance: beyond $2\sigma$ is fairly rare, beyond $3\sigma$ very rare.

Standardisation converts any normal value to a z-score: $$z = \frac{x - \mu}{\sigma},$$ the number of standard deviations $x$ lies from the mean. This maps any $N(\mu, \sigma^2)$ onto the standard normal $N(0, 1)$, so a single table (or function) handles all normal probabilities. A z-score of $+2$ means "two standard deviations above average" — a universal, unit-free way to compare values across different normal 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 ($z = \frac{x - \mu}{\sigma}$), not the value itself and not a probability. To get a probability you convert the z-score through the standard normal (table or function). Reading $z = 2$ as "probability 2" (impossible) or as the raw data value mistakes what standardisation produces.