In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is $${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}$$

The parameter ⁠${\displaystyle \mu }$⁠ is the mean or expectation of the distribution (and also its median and mode), while the parameter ${\textstyle \sigma ^{2}}$ is the variance. The standard deviation of the distribution is ⁠${\displaystyle \sigma }$⁠ (sigma). A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.

A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). (For other names, see Naming.)

The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.

The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when ${\textstyle \mu =0}$ and ${\textstyle \sigma ^{2}=1}$, and it is described by this probability density function (or density): $${\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}\,.}$$ The variable ⁠${\displaystyle z}$⁠ has a mean of 0 and a variance and standard deviation of 1. The density ${\textstyle \varphi (z)}$ has its peak ${\textstyle {\frac {1}{\sqrt {2\pi }}}}$ at ${\textstyle z=0}$ and inflection points at ${\textstyle z=+1}$ and ⁠${\displaystyle z=-1}$⁠.

Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss, for example, once defined the standard normal as $${\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}$$ which has a variance of ⁠${\displaystyle {\frac {1}{2}}}$⁠, and Stephen Stigler once defined the standard normal as $${\displaystyle \varphi (z)=e^{-\pi z^{2}},}$$ which has a simple functional form and a variance of ${\textstyle \sigma ^{2}={\frac {1}{2\pi }}.}$