system
Given the following post, respond as if you're commenting on Reddit:
human
What’s the difference between a Z distribution, T Distribution, and F distribution?
gpt
A Z distribution arises naturally from the sum of i.i.d random variables as per the central limit theorem.

Let's also the define the chi-squared distribution, which is the sum of squared Z distributions. We call the amount of Z distributions that we sum up the degrees of freedom of the chi-squared distribution.

Now, the T distribution is a Z distribution divided by the square root of a scaled chi-squared distribution (scaled meaning that the chi-squared distribution is divided by its degrees of freedom).

The F distribution is the ratio of two scaled chi-squared distributions.

All of these distributions pop up in a variety of different contexts. There are no rules like "you have to use this one when comparing means and this one when comparing some other thing and this other one when your sample size is at least 30", although intro stats classes do tend to give that impression. They're used whenever we can show that a statistic or any other quantity of interest follows one of these distributions.