Normal distribution

The normal or Gaussian probability distribution is actually a family of distributions of the same general form, differing only in their location and scale parameters, commonly called the mean and standard deviation.

The shape of a graph of the distribution consists of a central bulge centered on the mean with about 99.7% of the area under the density curve between the mean plus and minus three standard deviations. Its resemblance to the shape of a bell has led to the shape of the normal distribution being called the "bell curve".

The probability distribution function for the normal distribution with mean μ and standard deviation σ is

p(x) = exp(-(x-μ)²/2σ²) / (2π)^1/2σ

One reason that this distribution occurs so often in statistical work is the Central Limit Theorem. Simply stated, this theorem says that if you add up a lot of little things, the resulting distribution will resemble the normal distribution. More precisely: if you have n independent identically distributed random variables with mean 0 and standard deviation 1, then n^-1/2 times their sum converges in distribution to the normal distribution with mean 0 and standard deviation 1.

Beware! There are random variables which do not have both a mean and a standard deviation. (The Cauchy distribution is a famous example.) Sums of such unfriendly random variables need not tend to normality.

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