Paintings are said to be balanced even when most of the stuff/action is on one side of the canvas. What are we talking about anyway then?
Statistical distribution is concerned with the numbers and sizes of objects in a composition whereas balance is concerned with their placement in space.
The aforementioned distribution is distribution over a mathematical curve which represents our probabilistic expectation of finding an object of size "x" within the area of the canvas which in turn is guided by our experience in everyday life, i.e. the viewing of reality itself.
For instance, the probabilty of finding an object larger than the canvas is "0". It simply doesn't fit.
A composition is "balanced" when the array of objects lies in a natural (more or less random) placement. Thus, if everything is aligned like bricks in a wall, and those things are natural objects, then the composition is perceived a running counter to experience ... hence, unbalanced.
What we see in everyday natural situations is a couple of large objects, several somewhat smaller ones and a host of small things. Like the craters on the moon. There are plenty of small craters but only few very large ones.
The cratering on the moon is moderated by the propensity of a moon to split into parts if the impact "missile" is too large. Hence, moon cratering is "unbalanced" in principle. there aren't enough ~ 1/2 moon sized craters. But balance is restored if we view at a smaller scale (~ 1000 kilometers would do it).
A good example of an "unbalanced and poorly distributed" work of art is that of Hieronymous Bosch (Garden of Earthly Delights). We see numerous groupings of people in the process of being ... hmmmm ... mortified? Taken as a whole, the distribution doesn't reflect our expectation. It is therefore compositionally "unbalanced". However, if each grouping is looked at individually, our sense of balance is restored. And in fact, that is the way it was intended to be viewed ... one group at a time. So it is equivalent to many small paintings in one.