How come it is fairer than the arithmetic mean?
I would like to learn the pros and cons of each as well
The geometric mean is the "correct" mean for ratios, which prices are. It's the arithmetic mean on the log-scale:
Which chart looks more "correct"?
jerfelix is right of course: As a seller you're interested in the best "bid", and as a buyer you're interested in the best "ask". bitcoincharts.com has a calculator for larger orders, on the Market Depth page.