I often hear "dann treffen wir uns gegen 8 Uhr irgendwo". How precise is this? Is it ±5 min or ±10 min or does it depend on the situation?
Are there other words which means "approximately at eight", but are either less precise or more precise?
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It depends on the persons and situation.
In my opinion it is around ±5 min or ±10 min and it sounds to me if a group is meeting before 8 (maybe somewhere in the city or at an event) and the other person will join them later but without any fixed time
It means around, about or towards the time. It's a rough time with some minutes tolerance, not important after the mentioned time or before it. (as therealmarv has answered) it depends on the person.
Other words which means "approximately at eight" (less to more precise*):
If you mean the precise time you would say:
* : it's hard for the first five or so entries in the list to determine which one is more precise. At least colloquially, they can all mean the same.