Normal Distribution

The normal distribution (or Gaussian) is the iconic bell-shaped continuous probability distribution. Its density:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)$

is fully determined by two parameters: the mean $\mu$ (location) and standard deviation $\sigma$ (spread).

Key properties:

• Symmetric about $\mu$.
• 68-95-99.7 rule: $\approx 68\%$ of values within $1\sigma$, $95\%$ within $2\sigma$, $99.7\%$ within $3\sigma$.
• The standard normal $N(0, 1)$ is the canonical reference; any normal can be standardised via $z = (x - \mu)/\sigma$.

The normal arises everywhere because of the Central Limit Theorem: the sum of many independent random variables tends to normal regardless of their individual distributions. This makes it the default model for measurement errors, IQ, height, exam scores, and the foundation of confidence intervals, hypothesis tests, and Gaussian processes.