# Is there a minimum number of entries in a bin to consider it for computing $\chi^2$ ? and for gaussian behaviour? [duplicate]

Let's consider a set of data points of some observable in several bins, and a theoretical model. The agreement between the two is using the $$\chi^2/ndf$$ formula (where we divide $$\chi^2$$ by the number of freedom). Let's concentrate on the $$\chi^2$$ part :

The $$\chi^2$$ formula itself is $$\chi^2=\sum_i (d_i-t_i)^2/\sigma_i^2$$

where :

$$i$$ is the bin of a given observable

$$d_i$$ is data in the bin $$i$$

$$t_i$$ is the prediction from theory in the bin $$i$$

Is there a needed minimum number of entries from data in a given bin $$i$$ in order not bias completely the $$\chi^2$$, by making it exploding ?

Is there a connection with gaussian behaviour from Poisson distribution ?

• The simple answer is no because it isn't the actual bin counts that matter: it's their expected counts (which play the role of $\sigma_i^2$).
– whuber
Apr 2 '20 at 16:17
• @whuber : ok, sorry, is there a minimum of expected count for theory ? Apr 2 '20 at 21:54
• There is a rule of thumb frequently quoted: the minimum should be 5. However, that applies only when the number of bins is small (10 or less, roughly). As the number of bins grows, more exceptions can be accommodated. The full scope of applicability of the chi-squared distribution in the chi-squared test is thereby difficult to characterize.
– whuber
Apr 3 '20 at 11:36
• @whuber : thanks a lot Apr 3 '20 at 16:44