Extraordinary Claims Require Extraordinary Evidence

[Angrboda]

If you're talking about science, you don't need this phrase at all. Any claim must make specific testable and falsifiable predictions. This is where anything to do with religion falls down instantly because it makes no such things.

Actually, different branches of science, say medicine and physics, may use different standards.

It comes back to what I was saying about the statistics of type I and type II errors. Extraordinary claim means one where there is no plausible mechanism of action, or none at all. For example, ESP. Extraordinary evidence is quantified in terms of the statistical likelihood of getting the same results by chance instead of cause and effect. For ESP, you would use tighter standards.