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Normal equations

Two examples of the Normal (m,s) distribution (or in Crystal Ball the Normal(mean, standard deviation))are given below:


1. Modeling a naturally occurring variable

The Normal, or Gaussian, distribution occurs in a wide variety of applications due, in part, to Central Limit Theorem which makes it a good approximation to many other distributions.

It is frequently observed that variations of a naturally occurring variable are approximately Normally distributed: for example, the height of adult European males, arm span, etc.

Population data tend to approximately fit to a normal curve, but the data usually have a little more density in the tails.

2. Distribution of errors

A Normal distribution is frequently used in statistical theory for the distribution of errors (for example, in least squares regression analysis).

3. Approximation of uncertainty distribution

A basic rule of thumb in statistics is that the more data you have, the more the uncertainty distribution of the estimated parameter approaches a Normal. There are various ways of looking at it: from a Bayesian perspective, a Taylor series expansion of the posterior density is helpful; from a frequentist perspective, a Central limit Theorem argument is often appropriate: Binomial example; Poisson example.

4. Convenience distribution

The most common use of a Normal distribution is simply for its convenience. For example to add Normally distributed (uncorrelated and correlated) random variables, one combines the means and variances in simple ways to obtain another normal distribution.

Classical statistics has grown up concentrating on the Normal distribution, including trying to transform data so that they look Normal.  The Student-t distribution, and the Chi Squared distribution are based on a Normal assumption. It's the distribution we learn at college. But take care that when you select a Normal distribution it is not simply through lack of imagination: that you have a good reason for its selection, because there are many other distributions that may be far more appropriate.


Many distribution types converge to a Normal distribution as their coefficient of variability (i.e. the standard deviation divided by the mean) approaches zero including the Beta, Lognormal, Student-t, NegBinomial, Binomial, Gamma, Poisson and Chi Squared. Weibull(0,3.25,a) is very close to a Normal distribution too. Explanations of how the Normal distribution approximates other distributions is provided here.

The Normal distribution extends over the entire range of real numbers i.e. from -infinity to +infinity so it may sometimes be inappropriate to use it for variables like weight, length, time, etc. where a negative value is nonsensical. However, if the coefficient of variability is less than 1/3, there is at most a 0.15% chance of producing a negative value.

The Normal distribution is often called the Gaussian distribution after the German mathematician Carl Friedrich Gauss (1777-1855) and sometimes also called the Bell Shaped distribution.

In statistics books, the notation Normal   (m,s^2) instead of Normal(m,s) is sometimes used to ensure one does not mix up the standard deviation and the variance.

The Excel function NORMDIST(x,m,s,0) returns the probability density function for the Normal(m,s) distribution, while NORMDIST(x,m,s,1) returns its cumulative distribution function. The Excel functions NORMINV, NORMSDIST, and NORMSINV are also useful.

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