3.3: Geometric Mean
The mean is a measure of the central tendency of a data set. In some data sets, the data is inherently multiplicative, and the arithmetic mean is not useful. For example, the human population multiplies with time, and so does the credit amount of financial investment, as the interest compounds over successive time intervals.
In cases of multiplicative data, the geometric mean is used for statistical analysis. First, the product of all the elements is taken. Then, if there are n elements in the dataset, the nth root of the products is defined as the geometric mean of the data set. It can also be expressed via the use of the natural logarithmic function.
For example, suppose money compounds at annual interest rates of 10%, 5%, and 2%. In that case, the average growth factor can be calculated by computing the geometric mean of 1.10, 1.05, and 1.02. Its value comes out to be 1.056, which means that the average growth rate is 5.6% per annum.
It can be shown that the geometric mean of a sample data set is always quantitatively less than or at most equal to the arithmetic mean of the sample.