Normal Distribution
The normal distribution is a symmetric, bell-shaped probability distribution that describes many natural phenomena.
Formula
f(x) = \dfrac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}
Definition
The normal distribution is a bell-shaped curve where most values cluster near the mean and fewer values appear as you move farther away; it is perfectly symmetric. Formally, $N(\mu, \sigma^2)$ is a symmetric, unimodal, continuous probability distribution fully described by its mean $\mu$ and standard deviation $\sigma$, with PDF $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$; it is the maximum entropy distribution for a given mean and variance. The multivariate normal $N(\mu, \Sigma)$ generalizes it with mean vector $\mu$ and covariance matrix $\Sigma$, with PDF $f(x) = (2\pi)^{-k/2}|\Sigma|^{-1/2}\exp\left(-\tfrac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)$.
Example
Heights of adults, scores on standardized tests, and measurement errors tend to follow a normal distribution, with most people clustering around the average and very tall or very short people rarer. IQ scores are designed to be $N(100, 15^2)$: about $68\%$ of people score between $85$ and $115$, about $95\%$ between $70$ and $130$, and only about $0.3\%$ below $55$ or above $145$. The moment generating function of $N(\mu, \sigma^2)$ is $M(t) = \exp(\mu t + \sigma^2 t^2/2)$, which completely characterizes the distribution and whose log is a quadratic in $t$, a property unique to the normal among all distributions with finite moments.
Key Insight
The normal distribution is sometimes called the "bell curve" because of its shape, and nature produces it often because many things result from the sum of many small random effects, an empirical rule that says $68\%$ of data falls within $1\sigma$, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$ of the mean. The central limit theorem (CLT) states that if $X_1, \ldots, X_n$ are i.i.d. with mean $\mu$ and variance $\sigma^2$, then $\sqrt{n}(\bar{x}-\mu)/\sigma$ converges in distribution to $N(0,1)$, the mathematical foundation for normal-based inference and the reason for the normal distribution's ubiquity in statistical practice.