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

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SUMMARY

The discussion centers on the significance of the chi-squared statistic (\(\chi^2\)) divided by the number of degrees of freedom (ndf) being close to one, indicating a good fit in statistical modeling. The formula for \(\chi^2\) is presented as \(\chi^2 = \sum_{i=1}^{n} \frac{(X_i - \mu_i)^2}{\sigma_i^2}\), where \(X_i\) represents observed values, \(\mu_i\) are the expected values, and \(\sigma_i^2\) denotes the variances. A value close to one suggests that the model's estimates for \(\mu_i\) and \(\sigma_i^2\) are accurate, leading to a well-fitted model. Deviations above or below one indicate potential issues with the model fit.

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  • Understanding of chi-squared statistics
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  • Knowledge of normal distribution and variance
  • Basic competency in statistical modeling techniques
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DougUTPhy
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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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