Question

What are chi distributions, how do we use them, when do we use them, and why are they important?

Answer 1

Chi square distribution: The Chi-square distribution arises when we have a sum of squared normal distributed variables.

The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square. Two of the more common tests using the Chi Square distribution are tests of deviations of differences between theoretically expected and observed frequencies (one-way tables) and the relationship between categorical variables (contingency tables). Numerous other tests beyond the scope of this work are based on the Chi Square distribution

It is important because  important estimators such as the sample variance (if the random sample is i.i.d normal) and other functions like the deviance and log-likelihood ratio have sampling distribution equivalent or related to the chi-square distribution. This makes it important in statistical inference (obtaining confidence intervals or hypothesis testing)