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The $\chi^2$ statistic is independent of the number of degrees of freedom. But converting that statistic to some type of $p$-value does depend on the degrees of freedom. That is, you calculate $\chi^2$, then with that number and the degrees of freedom you look it up the $p$-value in a $\chi^2$ table. As for the "corrected" version of this test, you may find ...

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