# Normal distribution

Probability density function Probability density function for the normal distribution The green line is the standard normal distribution | |

Cumulative distribution function Cumulative distribution function for the normal distribution Colors match the image above | |

Parameters | <math>\mu</math> location (real) <math>\sigma^2>0</math> squared scale (real) |
---|---|

Support | <math>x \in\mathbb{R}\!</math> |

Probability density function (pdf) | <math>\frac1{\sigma\sqrt{2\pi |

Cumulative distribution function (cdf) | {{{cdf}}} |

Mean | {{{mean}}} |

Median | {{{median}}} |

Mode | {{{mode}}} |

Variance | {{{variance}}} |

Skewness | {{{skewness}}} |

Excess kurtosis | {{{kurtosis}}} |

Entropy | {{{entropy}}} |

Moment-generating function (mgf) | {{{mgf}}} |

Characteristic function | {{{char}}} |

cdf =<math>\frac12 \left(1 + \mathrm{erf}\,\frac{x-\mu}{\sigma\sqrt2}\right) \!</math>| mean =<math>\mu</math>| median =<math>\mu</math>| mode =<math>\mu</math>| variance =<math>\sigma^2</math>| skewness =0| kurtosis =0| entropy =<math>\ln\left(\sigma\sqrt{2\,\pi\,e}\right)\!</math>| mgf =<math>M_X(t)= \exp\left(\mu\,t+\frac{\sigma^2 t^2}{2}\right)</math>| char =<math>\chi_X(t)=\exp\left(\mu\,i\,t-\frac{\sigma^2 t^2}{2}\right)</math>|

}}
The **normal distribution** is a probability distribution. It is also called **Gaussian distribution** because it was discovered by Carl Friedrich Gauss.^{[1]} 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 (so-called 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.

## References

## Other websites

- Free Area Under the Normal Curve Calculator from Daniel Soper's
*Free Statistics Calculators*website. Computes the cumulative area under the normal curve (i.e., the cumulative probability), given a z-score. - Interactive Distribution Modeler (incl. Normal Distribution).
- GNU Scientific Library – Reference Manual – The Gaussian Distribution
- Normal Distribution Table
- Download free two-way normal distribution calculator
- Download free normal distribution fitting software