# definition of chi-square statistic

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?

• As currently posted, neither is a chi-squared statistic. numerator needs to be squared. – BruceET Apr 26 '19 at 2:09
• @BruceET I'm sorry that I missed it. The formulas are corrected now. – Nownuri Apr 26 '19 at 2:12
• 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. – whuber Apr 26 '19 at 12:44

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.