geometric mean

The geometric mean of a set of positive values is determined by taking the \(n\)th root of the product of the values, where \(n\) is the number (count) of the values. For example, the geometric mean of the numbers 9, 14, and 31 is given by

\[\sqrt[3]{(9\times 14\times 31)}\approx 15.75\].

The geometric mean is useful when we would like to know the average value of factors in a product, for example when calculating an average rate of return on an investment over a period when the rates have taken on several different values.

We can give the geometric mean of two values a physical interpretation by thinking of the values as the side lengths of a rectangle. The geometric mean of the values is then the side-length of a square having the same area.

Figure 1: Finding a Square of Equal Area