Education: Normal Distribution

Normal Distribution

Normal Distribution Definition

The normal distributions are a very important class of statistical distributions. All normal distributions are symmetric and have bell-shaped density curves with a single peak. A function that represents the distribution of many random variables as a symmetrical bell-shaped graph.

A normal distribution in a variate

with mean

and variance

is a statistic distribution with probability density function

(1)

on the domain

. While statisticians and mathematicians uniformly use the term "normal distribution" for this distribution, physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to it as the "bell curve." Feller (1968) uses the symbol

for

in the above equation, but then switches to

in Feller (1971).
de Moivre developed the normal distribution as an approximation to the binomial distribution, and it was subsequently used by Laplace in 1783 to study measurement errors and by Gauss in 1809 in the analysis of astronomical data (Havil 2003, p. 157).
The normal distribution is implemented in Mathematica as NormalDistribution[mu, sigma].
The so-called "standard normal distribution" is given by taking

and

in a general normal distribution. An arbitrary normal distribution can be converted to a standard normal distribution by changing variables to

, so

, yielding

(2)

The Fisher-Behrens problem is the determination of a test for the equality of means for two normal distributions with different variances.
The normal distribution function

gives the probability that a standard normal variate assumes a value in the interval

			(3)
			(4)

where erf is a function sometimes called the error function. Neither

nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated.

Normal Distribution Curve

The normal distribution is the limiting case of a discrete binomial distribution

as the sample size

becomes large, in which case

is normal with mean and variance

			(5)
			(6)

with

.
The distribution

is properly normalized since

(7)

The cumulative distribution function, which gives the probability that a variate will assume a value

, is then the integral of the normal distribution,

			(8)
			(9)
			(10)

where erf is the so-called error function.
Normal distributions have many convenient properties, so random variates with unknown distributions are often assumed to be normal, especially in physics and astronomy. Although this can be a dangerous assumption, it is often a good approximation due to a surprising result known as the central limit theorem. This theorem states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution. Many common attributes such as test scores, height, etc., follow roughly normal distributions, with few members at the high and low ends and many in the middle.
Because they occur so frequently, there is an unfortunate tendency to invoke normal distributions in situations where they may not be applicable. As Lippmann stated, "Everybody believes in the exponential law of errors: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation" (Whittaker and Robinson 1967, p. 179).
Among the amazing properties of the normal distribution are that the normal sum distribution and normal difference distribution obtained by respectively adding and subtracting variates