Accuracy is a fascination of mine.
The unrealistic hyping of accuracy is a pet annoyance of mine.

For the sake of this discussion, I
must assume the position of "Chief Egghead". I'll state
up front, there are probably plenty of readers who are better
qualified to quote the math, but I'll take a shot at it, and,
I think you might find a few things of interest.

Accuracy, in the shooting sports, is
identified in terms of MOA. That's "Minute of Angle".
It describes a measurement of a portion of the Arc (curve) of
a circle's circumference (outside rim). Take your group, as printed
on the target backer. Mentally extend a string from the impact
center of the two widest shots in your target's group, and run
those strings aaaaaaaaallllll the way back to the shooting bench
and down the barrel into the breech. That area inside the strings,
from the breech to the target, is called a "Sector".
It's a "slice of pie" from roughly the breech to the
target.

Let's talk about the breech, the muzzle,
and the target. Often discussions of this type talk only about
the line from muzzle to the target. Wrong. The muzzle can be almost
anywhere and still shoot a group onto the target. You need "three"
points in space to define accuracy potential. These are the pellet
in the breech, the muzzle position (which will direct the shot)
and the target which will receive the shot. The breech to muzzle
line determines where the barrel is pointing. Forget trajectory
for this egghead discussion. For this discussion - all rifles
shoot dead flat. Be clear - I KNOW rifles don't shoot perfectly
flat, but; unless you want to turn this completely into a NASA
trajectory paper, let's assume just this once, that they do.

Now, what's required to steer that
pellet into an accurate group? Just SIMPLY hold the segment of
the line from breech to muzzle within the designated mathematical
variation to achieve the desired group on the target. Oh yeah,
we'll have to assume the rifle is perfect, the pellets are perfect,
the shooter's hold is consistent, and a few other things. The
reason will very shortly become obvious.

So, just how much can we move the barrel
around and get a good group? Let's see. One MOA is always tossed
out as a good group - although many claim they can beat it on
any day of the week. So, what does the math say? In layman's terms,
most people use 1"@100 yards to describe one MOA. It's real
close to that. We'll do the math anyway and see. But, if 1"@100
yards is one MOA, then 3/4"@75 yards is also One MOA, and
1/2"@50 yards is also One MOA. Remember, we're working backward
toward the rifle, from the target, inside those strings we stretched
from the impact centers of the outside holes of our group. The
whole area inside the strings is called a "sector" remember.
But the strings themselves, are the outside lines of that sector
and the distance between them, at any point along the range from
target to breech, can be measured as to degrees of arc, and a
MOA measurement determined from it.

If we move the muzzle, we clearly alter
the direction of flight. How much matters? Let's see. For that
1/2"@50 yards that seems so commonly talked about today,
we must see what the "Arc" is for muzzle movement at
a relatively common barrel length. Let's just pick 18". Now
- this isn't a discussion of ideal barrel lengths. Forget that.
We need a number - so I've picked 18" as a typical barrel
length.

To get the Arc measurement of One MOA
at 18", we must use the following formula. AC=Pi times degree
of arc times radius divided by 180. Now, first we have another
problem ... Minute of Angle and Degree of Angle are two different
things. We need to convert MOA into degrees. Do this by multiplying
by .016667. OK - so One MOA is .016667 degrees. Having applied
the variables to the formula, we find that our One MOA group,
demands that we maintain our 18" barrel's breech/muzzle/target
alignment within .0052" at the muzzle. Read in English ...
that means we can toss the muzzle around anywhere we want ...
inside FIVE THOUSANDTHS of an inch!! What's that you say...? Five
thousandths of an inch - about the thickness of a sheet of letterhead
paper. O....K....

Now, it's simply a fact that, once
stuff starts going wrong, it's really a freak thing if several
errors work to correct and/or cancel themselves out. Things usually
go off the tracks quickly and error compounds error. Can you now
see why, for this discussion, we assumed all rifles shoot dead
flat? And why we needed to assume that the barrel was perfect,
power plant was perfect, the pellet was perfect, and that our
shooting technique remained perfectly consistent?

Now, that's One MOA for an 18"
barrel. That's the 1/2" @ 50 yards you hear people say they
shoot off their knee all day long. What's a 1/4" @ 50 yards
require? Just holding the breech/muzzle/target line to within
2.6 THOUSANDTHS OF AN INCH.... for each shot of the 3 or 5 shot
group. Piece of cake. Now, if the pellets vary any at all, or
if the rifle, barrel, anything, is less than perfect.... you are
toast. This is ONLY the mathematical absolute STARTING requirements
for these often discussed groups. This degree of perfection MUST
be in place AND any variables will only degrade the group size.

You have a 9" barrel on your springer?
Just divide the breech/muzzle/target line accuracy hold requirement
by 1/2.