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One of the reloading manuals - possibly Sierra? - mentions using 7 rounds as the sample size they use to be satisfied that they are able to predict "close enough". So, take it as a sampling exercise - whether "sighting in", whether "tightness of group", whether "velocity" - without shooting 5,000 rounds, you are trying to predict what the 5,000 group would look like.
If you have ever fired 10 shot groups at a target - say you get a 2" outside to outside grouping - now check how many "1/2" apart triples there are in there. So, did you get a "lucky" 3 shot group, or is that group of three shots actually representative of "real"??
In actual practice, I think I have shot enough over past 45 years to be able to accurately "call my shots". So, when working up a load or sighting in a scope, I want three solid "good shots". I make changes based on that. If my son is shooting, I want to see five shots - he simply does not put in the trigger time that he should. I also confirm results with several 3 shot groups - so my "sighting in" exercise often results in 3 x 3 shot groups, each on a different target - so, I suppose, that is really a 9 shot group. Makes me feel better about it...