Also known as the normal distribution or bell curve, is a continuous probability distribution that is widely used in statistics, mathematics, and the natural and social sciences. It is named after the German mathematician Gauss.

The Gaussian distribution is characterized by its symmetric bell-shaped curve, which is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The mean represents the center of the distribution, while the standard deviation determines the spread or dispersion of the data.

If a population has a normal distribution, then in a single trial the probability of getting an outcome $x$ is

$$ P(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{[(x-\mu)/\sigma]^2/2} $$

where $e$ is Euler's number. The distribution is shown below for three different mean and standard deviations.

The full width at half-maximum $\text{FWHM} = 2.354\sigma$ and probable error $\text{PE}=0.6745\sigma=0.2865\text{ FWHM}$. The standard deviation, FWHM and PE are shown below.