Mean, Geometric

The geometric mean is a measure of central tendency, and like all measures of central tendency, it is used to identify a single numerical value that is most “typical” or representative of a data set. The geometric mean is not to be confused with the much more prevalent arithmetic mean as they answer distinctly different questions about a data set. The arithmetic mean of a data set answers the question, “with what single number could each value in the data set be replaced without changing the sum?” However, the geometric mean answers the question, “with what single number could each value in the data set be replaced without changing the product?” Although it is of very limited use as a descriptive statistic, the geometric mean is very useful as an inferential statistic. This is especially true for research in fields such as finance, economics, and demographics wherein rates of growth and decline are both relevant and important. The geometric mean is calculated by multiplying all values in a data set, then finding the nth root where n is the number of values in that data set. It has a moderate bias toward smaller values in the data set, meaning that effect of large outliers on the value of the geometric mean is smaller than the effect of small outliers. For example, the [Page 936]data set {3, 6, 6, 6, 7, 9, 9, 12, 14} has a geometric mean of 7.34 because

$$\begin{array}{l}3\times 6\times 6\times 6\times 7\times 9\times 9\times 12\times 14=\mathrm{61,725,888}\\ \mathrm{Geometric}\mathrm{Mean}\mathrm{is}61,725,888\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.=7.34\end{array}$$

By comparison, that data set has an arithmetic mean of 8. Formally, the geometric mean, G, of a set of n numerical data {x1, x2, . . . , xn} is defined as

$$G=\mathrm{n}\sqrt{a_1\times a_2\times ...a_n}$$

where the operator Π is known as the product notation, and indicates that all values in the series should be multiplied together.