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I'm trying to find more information about this equation:

$\chi^2=\sum_{k=1}^N\big(\frac{f(x_k)-y_k}{\sigma_k}\big)^2$ for datapoints $(x_k, y_k, \sigma_k)$ and $\sigma_k$ is the uncertainty on $y_k$. I have been told this is a chi-square goodness of fit test for a fit $f(x)$. From this, I want to calculate a reduced chi-square test using $R\chi^2 = \frac{\chi^2}{N_{dof}}= \frac{\chi^2}{N_{pts}-N_{fit}}$ where $N_{pts}$ is the number of points and $N_{fit}$ is the number of degrees in freedom in the fit function. For example, a linear fit would have $N_{fit}=2$ since the equation dictating it is $f(x) = mx+b$. I have been trying to find more information about these equations and where they come from but cannot find them searching for chi-square and reduced chi-square. Are these what I think they are?

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Yes, this is a chi-squared statistic, but the terminology you are using is unusual in statistics and is probably from some other discipline. The unusual term reduced chi-square has a wikipedia page https://en.wikipedia.org/wiki/Reduced_chi-squared_statistic.

What you say is correct, you should probably read about nonlinear regression and goodness of fit there.

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