# Relating chi-squared and gaussian curves

1. Nov 26, 2009

### Ai52487963

Simple question: can a chi-squared be represented as a gaussian distribution? I'm wondering if I can take some chi-squared numbers that I have and represent them as increasing/decreasing widths of FWHM of a gaussian. Can I?

2. Nov 27, 2009

### SW VandeCarr

The Chi Sq distribution is the sum of the squares of a set of independent Gaussian RVs with mean 0 and unit variance. However, a generalized Chi Sq distribution can be derived from Gaussian RVs with non-zero means and non unit variance. This is the basis of Goodness of Fit tests. So the answer to your question is generally yes. Note you must distinguish between the Chi Sq distribution and Chi Sq tests which are highly dependent on how the data is categorized and on relative bin sizes.

Last edited: Nov 27, 2009
3. Nov 28, 2009

Suitably standardized, probabilities from a central chi-square distribution can be approximated by a normal distribution - the approximation improves as the number of degrees of freedom increases.

A similar result can hold for the non-central (I assume this is what SW meant be generalized) chi-square distribution.

However, each individual case must be investigated on its own; the problem is (obviously) that central chi-square distributions are right-skewed, and have a natural boundary at zero, while the normal distribution has neither of these properties.