Normal distribution

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In probability theory, the normal (or Gaussian) distribution, is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function is  bell”-shaped, and is known as the Gaussian function or bell curve:[nb 1]

where parameter μ is the mean (location of the peak) and σ 2 is the variance (the measure of the width of the distribution). The distribution with μ = 0 and σ 2 = 1 is called the standard normal.

The normal distribution is considered the most  basic” continuous probability distribution due to its role in the central limit theorem, and is one of the first continuous distributions taught in elementary statistics classes. Specifically, 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[1] 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.

Note that a normally-distributed variable has a symmetric distribution about its mean. Quantities that grow exponentially, such as prices, incomes or populations, are often skewed to the right, and hence may be better described by other distributions, such as the log-normal distribution or Pareto distribution.

In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavy-tailed distribution such as the Student’s t-distribution.

From a technical perspective, alternative characterizations are possible, for example:

  • The normal distribution is the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e. other than the mean and variance) are zero.
  • For a given mean and variance, the corresponding normal distribution is the continuous distribution with the maximum entropy.

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