Ask any shotgunner about the balance of his gun, and he will tell you it is lively, or, perhaps that it swings smoothly. Perhaps he does not particularly like the feel of the gun and he compares its dynamics to a pig on the end of a shovel.
Terms such as "Lively" or "Whippy" convey what the shooter is experiencing when he handles his gun. But, the terms are subjective.
It is possible, however, to measure certain gun properties that allow us to evaluate gun balance quantitatively.
This website gives details on the construction, calibration and use of a torsion pendulum, similar to that described by the well known British gun writer, Gough Thomas, in order to determine a gun's Moment of Inertia (MOI).
A gun's MOI is one of the properties that significantly affects a shooter's perception of the gun's dynamics.