It seems to me that there is no right number of shots for a group. Any number per group will give you information if you analyze the results correctly and are honest with yourself. Some like five shots. Some like ten shots. Some like Fords. Some like Chevys.
There IS, of course, such a thing as a lucky ten shot group you just have to shoot more groups to find one. Same with twenty shot groups or ---.
Whether shots per group make a difference depends on what you are trying to find.
If you are trying to determine what we usually call accuracy of the rifle/load, then:
If you shoot ENOUGH five shot groups and they average 1.0 MOA -AND the point of impact isn't shifting, you can predict that an average of enough ten shot groups will be about 1.24 MOA and an average of enough three shot groups will be about 0.81 MOA. That's just statistics.
The trick is shooting ENOUGH which we often don't have the patience to do. “Enough” is about the same number of total shots no matter how many shots are in the individual groups.
One problem with shooting three ten shot groups instead of six five shot groups is that a flier or two caused not by the rifle or load but by bad hold, wind, tired eyes etc. has a disproportionate effect on the average, and makes the rifle/load look worse than it really is.
If you are checking for round groups or what Joe Brennan calls “in control", then ten shots are better than five shots and twenty are better than ten.
I like Rick's method of shooting the first five shot group again while shooting the second one. I have also used the plotting method to see what the composite group looked like.
Of course if you are a gun writer testing and writing about a new rifle for one of the popular gun mags, then three shot groups are clearly best (two would be better yet).
Then you can show a picture of the best group found and use phrases like, “when I did my part” or “the rifle wanted to shoot” and other such horse hocky and the gun manufacturer will be happy and continue to advertise in you publication making your boss happy, thus three shots are best.
John