Chi Square Distribution

What is Chi-Squared Distribution?

The chi-squared distribution is frequently encountered when testing hypotheses about model parameters. It is also used when modelling variables that are always positive (e.g., the VIX Index). A chi-squared random variable is the sum of the squares of v independent standard normal random variables.
$ Y=\sum_{i=1}^{y}Zi^{2} $

It's important to remember that a chi-squared distribution has v degrees of freedom, which is a concept that comes up when working with models with k parameters and n data points. Because calculating model parameters need a minimum number of observations, degrees of freedom measure the quantity of data available to evaluate model parameters (e.g., k). The degree of freedom utilised in testing in many models is n — k.

PDF of Chi-Squared Distribution:

Why are Chi-Squared tests necessary?

Chi-square tests allow us to evaluate observed and expected frequencies objectively. It is not always feasible to judge whether they are "different enough" to be regarded statistically significant merely by looking at them. Hence, it plays a critical role in making statistical decisions about our estimations.