Editing Geometric mean (section)

===Average proportional growth rate===
{{Further|Compound annual growth rate}}
The geometric mean is more appropriate than the [[arithmetic mean]] for describing proportional growth, both [[exponential growth]] (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the [[compound annual growth rate]] (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

As an example, suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, for growth rates of 80%, 16.7% and 42.9% respectively. Using the [[arithmetic mean]] calculates a (linear) average growth of 46.5% (calculated by <math>(80% + 16.7% + 42.9%)\div 3</math>). However, when applied to the 100 orange starting yield, 46.5% annual growth results in 314 oranges after three years of growth, rather than the observed 300. The linear average overstates the rate of growth.

Instead, using the geometric mean, the average yearly growth is approximately 44.2% (calculated by <math>\sqrt[3]{1.80 \times 1.167 \times 1.429}</math>). Starting from a 100 orange yield with annual growth of 44.2% gives the expected 300 orange yield after three years.

In order to determine the average growth rate, it is not necessary to take the product of the measured growth rates at every step. Let the quantity be given as the sequence <math>a_0, a_1,..., a_n</math>, where <math>n</math> is the number of steps from the initial to final state. The growth rate between successive measurements <math>a_k</math> and <math>a_{k+1}</math> is <math>a_{k+1}/a_k</math>. The geometric mean of these growth rates is then just:
:<math>\left( \frac{a_1}{a_0} \frac{a_2}{a_1} \cdots \frac{a_n}{a_{n-1}} \right)^\frac{1}{n} = \left(\frac{a_n}{a_0}\right)^\frac{1}{n}.</math>