Chi-Squared distribution | Vose Software

Chi-Squared distribution

Format: ChiSq(n)


The Chi-Squared 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 in the following graph:


The sum of the squares of n unit-Normal distributions (i.e. Normal(0, 1)^2) is a ChiSq(n) distribution: so ChiSq(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 ChiSq distribution.

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 ChiSq test). The method attempts to make a ChiSq distributed statistic by taking the sum of squared errors, normalizing them to be N(0,1).

The ChiSq tests and statistics get over-used (especially the GOF statistic) because the normality assumption is often tenuous.


As n gets large, it is the sum of a large number of [N(0,1)^2] distributions and, through Central Limit Theorem, approximates a Normal distribution itself.

Sometimes written as c2(n). Also related to the Gamma distribution: Chisq(n) = Gamma(n/2, 2).

The distribution was first discovered by Friedrich Robert Helmert in 1876, then again by Karl Pearson in 1900.


ModelRisk functions added to Microsoft Excel for the Chi-Squared distribution

VoseChisq generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseChisqObject constructs a distribution object for this distribution.

VoseChisqProb returns the probability density or cumulative distribution function for this distribution.

VoseChisqProb10 returns the log10 of the probability density or cumulative distribution function.

VoseChisqFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseChisqFitObject constructs a distribution object of this distribution fitted to data.

VoseChisqFitP returns the parameters of this distribution fitted to data.


Chi-Squared distribution equations