In probability theory and statistics, the normal distribution, or Gaussian distribution, is an absolutely continuous probability distribution with zero cumulants of all orders above two. The graph of the associated probability density function is “bell”-shaped, with peak at the mean, and is known as the Gaussian function or bell curve:

   f(x) = \tfrac{1}{\sqrt{2\pi\sigma^2}}\; e^{ -\frac{(x-\mu)^2}{2\sigma^2} },

where parameters ? and ??2 are the mean and the variance. The distribution with ? = 0 and ??2 = 1 is called standard normal.

The normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches (1.8 m). Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used.

By the central limit theorem, under certain conditions the sum of a number of random variables with finite means and variances approaches a normal distribution as the number of variables increases. For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural sciences, and social sciences as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption.

The Gaussian distribution was named after Carl Friedrich Gauss, who introduced it in 1809 as a way of rationalizing the method of least squares. One year later Laplace proved the first version of the central limit theorem, demonstrating that the normal distribution occurs as a limit of arithmetic means of any random variables. For this reason the normal distribution is sometimes called Laplacian, especially in French-speaking countries.
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