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Would it wrong to say that the definition of null hypothesis is as follows:

Null hypothesis is a statistical hypothesis that usually asserts that nothing special is happening with respect characteristic of the population.

Someone mentioned that this definition will not hold in some cases. Is it true that the above mentioned definition is incorrect is some cases? If so, can you mention a case where it would fail.

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    $\begingroup$ This is all very ambiguous, can you give concrete examples/references/quotes? $\endgroup$
    – user2974951
    Jul 1 at 10:00
  • $\begingroup$ Being pedantic about the English, if the definition says that something “usually” holds, then it is completely acceptable that there are some exceptions. It’s usually true that a high school basketball player won’t play professional basketball. Alas, there are players in the NBA. $\endgroup$
    – Dave
    Jul 1 at 10:23
  • $\begingroup$ "Nothing special is happening" might be useful for suggesting the intended meaning, but it certainly is no definition. When you think about it, it's hard to find anything objective in it whatsoever. Indeed, what does it even mean? In a statistical test or popularion, to what does "happening" refer and how do you characterize or quantify "special" (which sounds completely subjective and arbitrary)? $\endgroup$
    – whuber ♦
    Jul 1 at 13:49

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Definitions are not just characterised or judged by their correctness/accuracy. They need to be as precise as possible to be maximally useful. The problem with your definition is that it is vague in multiple respects, and arguably self-contradictory. 'usually asserts' and 'nothing special' are ambiguous and imprecise. Even worse, usually implies exceptions, which essentially undermines the whole definition - are there null hypotheses that this does not apply to? What characterises/defines those? And 'happening' is both ambiguous and irrelevant to many possible applications of the term.

It can be helpful for our own understanding to come up with loose explanations of terms in simple language that we can grapple with. But this is a step towards deeper understanding, and we should not mistake it for a rigorous definition.

As a mild caveat to what I wrote above, many fields do use definitions (or 'working definitions') that do have exceptions. Biology famously still doesn't have an unambiguous and universally applicable definition of what a 'species' is. But that's in large part because a 'species' is a construct that maps only loosely onto reality. It is an attempt to put hard boundaries on reality when reality itself is more porous (as the phenomenon of fertile hybrids of different species shows). This looseness is much less justifiable when dealing with mathematical (and most statistical) constructs. Where greater precision is possible, we should aim for it.

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