Skip to content

Normal Distribution

Definition

Let μ∈R\mu\in\mathbb{R} and σ>0\sigma>0. A continuous random variable XX has a normal distribution with mean μ\mu and variance σ2\sigma^2, written X∼N(μ,σ2)X\sim\mathcal{N}(\mu,\sigma^2), if its density is

fX(x)=1σ2πexp⁡(−(x−μ)22σ2).f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).

Interpretation

A normal distribution is the familiar bell-shaped distribution. It is widely used because many sums and averages of independent variables behave approximately normally, by the central limit theorem.

A bell-shaped normal density centred at mu, with the area within one standard deviation of mu shaded, about 68 percent.

The curve is symmetric about its mean μ\mu, and its variance is σ2\sigma^2. Approximately 68%68\% of its probability lies within μ±σ\mu\pm\sigma, and 95%95\% lies within μ±2σ\mu\pm2\sigma.

Move μ\mu to shift the bell and σ\sigma to widen it. The interval (μ−σ,μ+σ)(\mu-\sigma,\mu+\sigma) always holds about 68%68\% of the probability.

As a continuous random variable, the normal variable assigns probabilities by areas under the density. The exponential function applied to the negative squared standardized deviation gives its bell-shaped decay.

Key Results

  • A normal distribution is determined by its mean and variance.
  • Standardizing a normal random variable gives a standard normal random variable.
  • Sums of independent normal random variables are normal.

Examples

  • Measurement error is often approximated by a normal distribution.
  • The average of many small independent effects is often modelled as normal.
  • Manufacturing deviations from a target dimension are frequently modelled with a normal distribution.

Prerequisites

Sources

Aliases: Gaussian distribution, normal random variable, Gaussian random variable, normally distributed Keywords: bell curve, N(mu, sigma^2) Symbols: element of, mu, pi, real numbers, sigma, square root

Last updated: 20 October 2026 03:02