Normal Distribution

The Normal Distribution is one of the core elements of statistics.  Not only is it extensively used in modelling, it also forms the basis of much statistical theory.  It is one of the first distributions to acquire a formal definition, originating from the work of Abraham de Moivre in the early 18th Century.

Applications

It is used in scientific and technical applications where the variation in the observations is due to a single cause.  A common use is the distribution of measurement errors in experimental results and the variation of component dimensions in manufacturing processes.

Under certain conditions, the normal distribution is used as an approximation for other distributions, such as the binomial and Poisson.

Central Limit Theorem

One expression of the central limit theorem relates to the distribution of the mean of samples drawn from populations which are not necessarily themselves normally distributed.  As the size of the sample increases, the distribution of the sample means will tend towards normal.  This makes the normal distribution important in a wide range of measurement processes, this is expressed in the standard error of the mean.

Profile

Parameters

Parameter	Description	Characteristics
mean	Population mean	A float > -∞ and < ∞
stdev	Standard Deviation of Population	A float > 0

Range

The range of the normal distribution is from -infinity to +infinity, however, the practical limits are 3.9 standard deviations away from the mean.

Functions

Where erf(x) is the error function and Φ(x) is the Laplace function.

One method of obtaining an inverse probability function (i.e. g(p)) is to use an equation solving technique such a successive bisection in conjunction with f(x) to iterate to a value of x which corresponds to the required value of p.

Properties

Example

The example is based on a sample of pieces of wood cut into 20mm dowels by a fitter:

The length of the dowels is normally distributed and the curve fit is significant at the 10% level, suggesting that it can be used as a production monitoring device.

Random Number Generation

The lack of an inverse probability function means that the transform method of generating non-uniform random numbers from a standard uniform one is not efficient.  A common method of generating random numbers from a standard normal distribution is the Box-Muller algorithm which generates pairs of standard normal random numbers (r₁ and r₂) from two standard uniform random numbers (u₁ and u₂):

Basic style psuedo code for a single standard normal random number is shown below:

r1=Sqr(-2 * Log(rnd())) * Sin(2 * PI * rnd())

The standard normal number can be scaled for given values of mean and standard deviation with this addition:

r1=mean+r1*stdev

A common refinement is to generate random numbers in pairs and to use a flag to determine if a new pair should be generated, or a previous value used.

Parameter Estimation

The parameters of the normal distribution are the mean and standard deviation.  The matching moment and maximum likelihood equations are the same:

Page updated: 24-Nov-2004

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