ChiSquared(n) - No Crystal Ball distribution = Gamma(0,2,n/2)

The ChiSquared distribution is a right-skewed distribution bounded at zero. n is called the 'degrees of freedom' from its use in statistics below. Examples of the Chi Squared distribution are given below:

#### Uses

The sum of the squares of *n* unit-Normal distributions (i.e. Normal(0, 1)^2) is a Chisquared(*n*) distribution: so ChiSquared(2) = Normal(0,1)^2+Normal(0,1)^2 for example. It is this property that makes it very useful in statistics, particularly classical statistics.

In statistics, we collect a set of observations and from calculating some sample statistics (the mean, variance, etc) attempt to infer something about the stochastic process from which the data came. If the samples are from a Normally distributed population, then the sample variance is a random variable that is a shifted, re-scaled ChiSquared distribution: Example 1, Example 2.

The Chi Squared distribution is also used to determine the goodness of fit (GOF) of a distribution to a histogram of the available data (a ChiSquared test). The method attempts to make a ChiSquared distributed statistic by taking the sum of squared errors, normalizing them to be N(0,1).

In our view, the ChiSquared tests and statistics get over-used (especially the GOF statistic) because the Normality assumption is often tenuous.

### Generation

The Chi Squared distribution is not directly available in Crystal Ball. However, because of its relationship to the Gamma distribution we can constructed a Chi Squared distribution by using the following equation:

#### Chisquared(*n*) = Gamma(0, *n*/2, 2)

#### Comments

As *n* gets large, so it is the sum of a large number of [N(0,10^2] distributions and, through Central Limit Theorem, approximates a Normal distribution itself.

Sometimes written as c2(*n*).

The Excel function CHIDIST(x,*n*) returns the cumulative probability from a Chisquared(*n*) distribution.