If you calculate the average billing rate increase of the two law firms you paid the most during 2009 and 2010, and then divide the total of those increases by two, you have the arithmetic mean of those increases. So, Firm A increased 4 percent and firm B increased 6 percent. The arithmetic mean was 5 percent, (4+6)/2.
The geometric mean is more appropriate than the arithmetic mean for describing proportional growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of the increases requires a different calculation. In the example, the geometric mean is the square root of each firm’s change multiplied together.
The geometric mean, however, is 4 times 6 raised to the ½ power (since you have two numbers and the exponent is 1/n), which is 4.9 percent (it is always less than the arithmetic mean). In the US the Labor Bureau calculates the Consumer Price Index with geometric means. More generally, the geometric mean of n quantities is the nth root of their product, according to John D. Barrow, 100 Essential Things You Didn’t Know You Didn’t Know: Math Explains Your World (Norton 2008) at 236. So, if you had changes in billing rates for five law firms, you would multiply all five of their percentage changes and raise it to the 1/5 (.2) power (the fifth root).