# What can a chi-squared distribution be used to describe?

Dec 8, 2017

Chi Squared distributions can be used to describe statistical quantities which are a function of a sum of squares.

#### Explanation:

The Chi Squared distribution is the distribution of a value which is the sum of squares of $k$ normally distributed random variables.

$Q = {\sum}_{i = 1}^{k} {Z}_{i}^{2}$

The PDF of the Chi Squared distribution is given by:

f(x;k)=1/(2^(k/2)Gamma(k/2)) x^(k/2-1)e^(-x/2)

Where $k$ is the number of degrees of freedom, and $x$ is the value of $Q$ for which we seek the probability.

The usefulness of the Chi Squared distribution is in modelling things which involve the sums of squared values. Two specific examples are:

• Analysis of Variance tests (variance is a sum of squared values)
• Goodness of fit (for a least squares fit where the error is a sum of squared values)