# Degrees of Freedom help

1. Oct 17, 2011

### cahill8

I'm trying to compare fits to data in a certain way. I start with the initial data and fit and compute the $\chi$^2 value for example:

x: 1 2 3 4
y: 2.1 3.9 6.0 10.1

I get the linear fit, $\chi$^2 value and degrees of freedom=3.

Now with what I'm working with, I don't actually know the values of x. The above values of x are just assumed (but the y values are known). I'm looking to reduce the $\chi$^2 value so I let the fourth x value (4) equal 5 instead and then make a new linear fit with a new $\chi$^2 value. It's obvious that the new $\chi$^2 value will be smaller than the original but do I lose a degree of freedom by choosing the last x value to be 5 instead of assuming its 4?

I want to be able to compare the $\chi$^2/(degrees of freedom) value for the original case (final x=4) and the adjusted case (final x=5).

I'm confused for either case. E.g. If the degrees of freedom remain the same, I can keep adding more and more points to get a better fit without any penalty.
On the other hand, if the degrees of freedom are reduced, if I were to add more points, would the degrees of freedom eventually hit 0?