in statistics, can some measure like middle or median be kind of "proved" to be a good indicator of something in a situation, if you cannot come up with a counterexample which clearly shows that it's a bad indicator?
something like:
conjecture: let $x_1,x_2,...,x_n$ be the incomes of some n ppl. Then $mean({x_i})$ is a good indicator for what income the ppl in ${x_i}$ are getting.
counterexample:
let there be 10 ppl with 1k income and 1 with 1 billion, then mean is like 90 million or something, but clearly these 11 ppl are not millionares. QED.
Then if you couldnt come up with a counterexample like that, then mean would be a good choice here?
Before you can say that a particular measure is "good", you need to define what "good" means. Or alternatively, you need to define what makes a measure "bad" and try to minimise that "badness". Either way, that comes down to understanding what question you're trying to answer. If I'm trying to answer something about how much money everyone would have if it were split evenly between them, then the mean is a perfectly fine measure. On the other hand, if I'm going to pick a random person and I want to be able to guess their income as closely as possible, then depending on how much I penalise being wrong it's likely that the median or mode is a better choice.
It's well known that the mean is sensitive to outliers, meaning that if most of the values are clustered together but one or two values are very far away then they will pull the mean towards them in ways that won't affect the median or mode.