What are some real-world applications where I can leverage Chebychev's inqueality? All the examples I find are either related to coin tosses or some school scores related problems. Are there any slightly-more complex use cases where this inequality "saves" the day?
I once convinced a very tall woman to have dinner with me by using Chebychev's inequality to argue that her high partner height threshold would lead to many lonely nights spent with her cat. Alcohol was involved, so arithmetic mistakes were made that exaggerated the main thrust of the conclusion.
It didn't last.
In this case, the three-sigma rule would have served me better since heights are actually normal and the rule gives a tighter bound. Now, if she had an income threshold, this would have been a better anecdote.
While it's most often used for establishing bounds in various things, here's an example of it being used for constructing intervals for a real problem:
(It's not actually necessary for this problem; tighter bounds can be obtained.)