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The chi-squared statistic is defined as \begin{equation} \chi^{2} = \sum_{i=1}^{N} \frac{(O_{i} - E_{i})^{2}}{E_{i}} \end{equation} The $O_{i}$ is observed value and $E_{i}$ is expected value.

But it seems that sometimes the statistic is defined differently as \begin{equation} \chi^{2} = \sum_{i=1}^{N} \frac{(O_{i} - E_{i})^{2}}{{\sigma_{i}}^{2}} \end{equation} The $\sigma_{i}^{2}$ is variance. I found this definition in this Wikipedia page for reduced chi-square

Are both of them called "chi-squared statistic"? When do you need the former and when do you need the latter?

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    $\begingroup$ As currently posted, neither is a chi-squared statistic. numerator needs to be squared. $\endgroup$ – BruceET Apr 26 at 2:09
  • $\begingroup$ @BruceET I'm sorry that I missed it. The formulas are corrected now. $\endgroup$ – Nownuri Apr 26 at 2:12
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    $\begingroup$ Both are chi-squared statistics. It is important for readers to consult that Wikipedia page on "reduced chi-square" because it uses nonstandard notation: "$\sigma_i^2$" is an (unbiased) estimate of variance, not a model parameter. $\endgroup$ – whuber Apr 26 at 12:44
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To my understanding there are different types of chi-squared statistic. You can use the first formula to calculate the chi-squared test statistic as in Pearson's test independence while the second formula is applied in goodness of fit statistic tests. So it depends on the situation what formula you use.

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    $\begingroup$ In light of the edit that was made to the question, this answer is not correct: it will help to read the referenced Wikipedia article, which makes it clear the second statistic is the one that appears in goodness-of-fit tests. $\endgroup$ – whuber Apr 26 at 12:44
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    $\begingroup$ Thank you. I adjusted the answer to the edit in the question. $\endgroup$ – stats.and.r Apr 26 at 13:29
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The latter definition is the general definition of the chi-square statistic. The former definition is for a special case where the parameter $\theta$ is Poisson distributed. Note that in a Poisson distribution expected value is the same as variance.

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