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

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A chi-squared value divided by the number of degrees of freedom (\chi^2/ndf) close to one indicates a good fit because it suggests that the observed data closely matches the expected model. This statistic is derived from the sum of squares of standardized normal random variables, where each term represents the deviation of observed values from the model, normalized by the variance. When the model parameters are accurately estimated, each fraction in the sum approaches one, leading to \chi^2/ndf being near one. Values significantly above or below one can indicate poor fits, with higher values suggesting underestimation of variability and lower values indicating overfitting. Understanding this relationship is crucial for evaluating the quality of statistical models in experimental data analysis.
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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!
 
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The statistic has the form
<br /> \chi^2 = \sum_{i=1}^{n} \frac{(X_i - \mu_i)^2}{\sigma_i^2},<br />
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.
 

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