Errata....

On Dec 5, 9:14 am, joeu2004 I wrote:
You don't need to be a mathematician. Simply experiment with smaller
groups of simple numbers, e.g. 2 through 11. Try splitting into 2
groups of 5. Then try splitting into groups of 4 and 6.

You should discover that the geomean of several geomeans equals the
single geomean only when all data sets are the same size.

What I should have said is: you should discover that the geomean of
geomeans does not equal the single geomean when the data sets have
different sizes.

The conclusion that it works for same-sized data sets does require an
inductive proof. It might work for 2 equal groups by accident. But
you might get pretty convincing evidence (or not) by experimenting
with equal groups of 2, 3, and 5, for example.