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

  1. Mar 23, 2012 #1
    Why is [itex]\chi^2 / \mathrm{ndf}[/itex] (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 [itex]\chi ^2[/itex] -- 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. jcsd
  3. Mar 24, 2012 #2
    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. [itex]\mu_i[/itex] and [itex]\sigma_i^2[/itex] are well estimated, you suppose each fraction to be close to one. Hence the sum gives [itex]n[/itex] and therefore [itex]\chi^2/n[/itex] gives a number close to 1.
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