A regular average would suggest that the basket of goods used to compare from interval to interval does not change. In other words, a 1lb ribeye in 1950 is the same as a 1lb ribeye today (in quantity, not price). If it's included in the CPI then the price increase is reflected properly.
Using a geometric average implies that the basket of goods selected CHANGES depending on the change in prices. So in 1950 when a ribeye was cheap, 1lb of it was included in the basket of goods. Now that it's more expensive, only 0.5lbs of it are included in the basket.
This is a made up example. Prices are not correct. Just for illustration.
1950:
hamburger - $0.50/lb
ribeye - $2/lb
arithmetic mean: (2 + 0.5) / 2 = $1.25
geometric mean: sqrt(20.5) = $1.00
2012:
hamburger: $3/lb
ribeye: $18/lb
arithmetic mean: (3 + 18)/2 = $10.50
geometric mean: sqrt(318) = $7.34
Arithmetic inflation: 10.50/1.25 = 8.4
Geometric inflation: 7.34/1 = 7.3
The geometric mean has two wonderful properties:
1. It makes the average price lower by adjusting the quantities included based on their relative prices
2. It makes the inflation over time seem lower as well
I'm not saying that this is some giant conspiracy by "the man" to "keep us down" but I would argue that it's intellectually dishonest.
In order for the CPI to be useful to make comparisons from one year to another the quantities of goods in the basket have to remain constant. Since the geometric mean effectively varies the quantity in the basket (relative to arithmetic) I would argue that it robs the CPI of any power to provide real comparisons.