The OP is 99.99999% correct.
A 95% confidence interval of an actual ES or SD based on three shots would be quite broad.
I often seen people trying to tune 1” guns based on a handful of shots. If a gun, and brass(which most hunters don’t uniform or sort) and bullets aren’t all capable of shooting around .5MOA or less, then tuning becomes an long painful process that just about isn’t worth doing. When a rifle is good for .1MOA or less, and so is the brass(that was good to begin with and better after you uniformed and sorted it) and so are the bullets(go ahead and tell me that your Noslers are that good), and your dies are custom made, and your barrel is hand lapped(so that you know fouling hasn’t impacted your groups), and you shot ten consecutive 5-shot groups averaging .25MOA or less last weekend at the match, and you have 2-3 wind flags between you and the target, then seeing one round 5-shot group that is .250” and one oval 5-shot group that is .450” does actually tell you something. On the other hand, taking your Winchester brass and Nosler bullets out to the range for your factory gun, and loading with Lee or RCBS dies, shooting two or three 3-shot groups, and declaring that the .5” load was better than the 1” load, is almost impossible to prove statistically. The truth is, the .5” load and 1” load may be equal, or the 1” load might even be better.
. Every data point has some inherent value and does tell you something - depending on your needs your "n" may need to be big or it may be ok to be small.
