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Here's a neat article about tuned mass dampeners being used to balance wobbly skyscrapers. First off: that ball doesn't look so big to me.
Second, perhaps a similar effect contributes to womens' (anecdotally supported, at least) superior sense of balance.
(HT: GeekPress.)
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If women have better balance, maybe it's because they have lower centers of gravity. More weight in the hips and less in the shoulders.
Yeah, that diagram shows it to be about one and a half stories in diameter, which maybe seems a bit small, but it could just show that solid things don't need to be so big to be heavy. IIRC just one meter cubed of steel is about 8 tons. For 728 tons that would make the volume about 90m^3, which would make the radius about 3 meters.
I strongly doubt that women can balance well because of a massive ball inside of them. If anything, that would make a girl more likely to fall over. You should know this, being a prospective dad now. (many congratulations to both of you!)
There are various balancing tricks that are easy for women due to the hip vs shoulder weight distribution, such as picking up a bottle with your mouth from a crouching position, but I don't know that that translates to better balance in general.
You guys are hilarious :)