Probability density function The green line is the standard normal distribution 

Cumulative distribution function Colors match the image above 

Parameters  location (real) squared scale (real) 

Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  
Mean  
Median  
Mode  
Variance  
Skewness  0 
Excess kurtosis  0 
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
The normal distribution is a probability distribution. It is also called Gaussian distribution because it was discovered by Carl Friedrich Gauss. The normal distribution is a continuous probability distribution. It is very important in many fields of science. Normal distributions are a family of distributions of the same general form. These distributions differ in their location and scale parameters: the mean ("average") of the distribution defines its location, and the standard deviation ("variability") defines the scale.
The standard normal distribution (also known as the Z distribution) is the normal distribution with a mean of zero and a variance of one (the green curves in the plots to the right). It is often called the bell curve because the graph of its probability density looks like a bell.
Many values follow a normal distribution. This is because of the central limit theorem, which says that if an event is the sum of other random events, it will be normally distributed. Some examples include:
 Measurement errors
 Light intensity (socalled Gaussian beams, as in laser light)
 Intelligence is probably normally distributed. There is a problem with accurately defining or measuring it, though.
 Insurance companies use normal distributions to model certain average cases.
Images for kids

The ground state of a quantum harmonic oscillator has the Gaussian distribution.

The bean machine, a device invented by Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.

Marquis de Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.