Normal Distribution - Mathematical Background

The shape of the normal distribution resembles that of a bell, so it sometimes is referred to as the "bell curve", an example of which follows:


The normal distribution often is used to describe random variables, especially those having symmetrical, unimodal distributions. In many cases however, the normal distribution is only a rough approximation of the actual distribution. For example, the physical length of a component cannot be negative, but the normal distribution extends indefinitely in both the positive and negative directions. Nonetheless, the resulting errors may be negligible or within acceptable limits, allowing one to solve problems with sufficient accuracy by assuming a normal distribution.

The normal distribution can be completely specified by two parameters:

  • mean
  • standard deviation

If the mean and standard deviation are known, then one essentially knows as much as if one had access to every point in the data set. The so-called "standard normal distribution" is given by taking mu==0 and sigma^2==1 in a general normal distribution.

A normal distribution in a variant X with mean mu and variance sigma^2 is a statistic distribution with probability function

P(x)==1/(sigmasqrt(2pi))e^(-(x-mu)^2/(2sigma^2))   on the domain x in (-infty,infty)

The empirical rule is a handy quick estimate of the spread of the data given the mean and standard deviation of a data set that follows the normal distribution.

The empirical rule states that for a normal distribution:

  • 68% of the data will fall within 1 standard deviation of the mean
  • 95% of the data will fall within 2 standard deviations of the mean
  • Almost all (99.7%) of the data will fall within 3 standard deviations of the mean



The formula for variance is given as


Standard deviation is defined as square root of the variance.

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