# How should I interpret a Chi-Squared Result?

I've got a Model A with a reduced chi-square of 1.28. I've got a Model B with a reduced chi-square of 0.70. Which is a better model? The model closest to 1 or the model closest to zero?

(Yes, I know this is probably better on the math site, but I got no answer there. Besides, the chi-square is ubiquitous in astronomy and it would be useful to have a definitive answer so we can read the papers and understand the results)

• Would this be better taken to Cross Validated? Most physicists have a rather hit-n-miss training in statistical methods and interpretation. Commented Nov 30, 2015 at 21:59
• If most physicists have a poor training in statistical interpretation, don't you think it would be a good idea to fix that in the place they're most likely to look for information?
– user32023
Commented Nov 30, 2015 at 22:20
• I didn't say they are bad at stats, I said they have a spotty understanding. They generally know the right or standard approaches to use in the problems they work on, but don't necessarily have a wide spread understanding. Commented Nov 30, 2015 at 22:34
• You got no answer where? Commented Dec 1, 2015 at 0:09

The one closest to zero is the best fit, but depending on the conditions you can't rule out the model with 1.28. Most often you cannot rule out anything where Chi-squared is closer than 1 to the value of your best fit - but it does depend in reality on a bunch of things including the number of variables you used for your model fit, for example.

Numerical Recipes has a good description of Chi-squared fitting.

0.7 and 1.28 are both reasonnably close to zero (and to 1) and the Chi-squared test indicates that both are reasonnable fits. --- if one fit gave 0.7 and another gave 325.6 then you could rule out the second model. Normally for 68% confidence we allow chi-squared to increase by 1, but here with several paramters we may need to increase by more than 1 - so here the Chi-squared test says that both are plausible fits to the data

• A good fit and a good model are not the same thing. In particular you can always fit a dataset exactly with a high degree polynomial, but it generally won't have any physics content. Commented Nov 30, 2015 at 21:59
• @dmckee - yes that is a very good point - Chi-squared only gives the best fit and does not judge on the correctness of the model. - I put 'correct' at first then editted to 'best fit'.
– tom
Commented Nov 30, 2015 at 22:00
• I don't want to 'rule any model out'. I want to know which model is a better fit to the data. Model A has three fitting parameters, Model B has two. There are roughly 500 data points. I want to draw a conclusion in my paper. Is there a simple answer?
– user32023
Commented Nov 30, 2015 at 22:18
• "Model A has two fitting parameters, Model B has three. There are roughly 500 data points. Is there a simple answer?" Either fit is defensible on that basic, but you should have included that data in your question. Commented Nov 30, 2015 at 22:36
• So you're saying, what, exactly? That below a certain point in the chi-square test there is no objective criteria for comparing one model to another?
– user32023
Commented Nov 30, 2015 at 22:51