The Chi Squared statistic is used in 3 cases:

Test of adjustment

It allows to test if one distribution or more is adjusted to other distribution.

This is a non parametric test, meaning that it can be used even with non normal distributions.

For instance, you can check if one distribution is similar to a normal distribution.

 

\chi ^{2}=\sum \frac{\left ( O-E \right )^{2}}{E}

 

Test of variance

You can used the chi square to test the confidence interval of the variance of a normal distribution

 

Multiple test of adjustment

You can check among different groups if they are similar or different