Why is chi^2/ndf close to 1 a good fit?

1. Mar 23, 2012

DougUTPhy

Why is $\chi^2 / \mathrm{ndf}$ (number of degrees of freedom) close to one mean that a fit is a good fit?

I have had this question for a long time, and now I'm currently in a lab where the instructor and TA's love to see you talk about $\chi ^2$ -- so it's killing me! All I have ever heard is that it is a good fit, but I have never heard why. Or what the difference is between a being a little above or a little below one.

I hope math is a good board to put this in, I kind of feel like it's a statistics question.
Just a general question to quench my curiosity...
Thanks for any insight!

2. Mar 24, 2012

camillio

The statistic has the form
$$\chi^2 = \sum_{i=1}^{n} \frac{(X_i - \mu_i)^2}{\sigma_i^2},$$
i.e., it is a sum of squares of standardized normal random variables. If your fit is good, i.e. $\mu_i$ and $\sigma_i^2$ are well estimated, you suppose each fraction to be close to one. Hence the sum gives $n$ and therefore $\chi^2/n$ gives a number close to 1.