When a data set contains a large number of repeated values, you can simplify the process of computing the mean by using weights — the frequencies of a value in a sample or a population. You can then compute the geometric mean as a weighted average.

You can calculate the weighted geometric mean in the same way for both samples and populations. The formula is:

Here's the breakdown of this equation:

You apply an exponent to each element in the data set that equals the weight of the element. You then multiply these values together and raise to a power equal to one divided by the sum of the weights.

An exponent is the superscript in an expression such as 34; in this case, the base is 3 and the exponent is 4. This is shorthand for multiplying 3 by itself four times:

Note that in many formulas and Microsoft Excel, the asterisk (*) represents multiplication. In Excel the carat (^) represents exponentiation.

As an example, a marketing firm conducts a survey of 20 households to determine the average number of cellphones each household owns. Here's the sample data from this survey:

Number of Cell Phones Per Household Number of Households
1 2
2 5
3 6
4 4
5 3

To figure out the weighted geometric mean, follow these steps:

1. Compute the value of each Xi with an exponent equal to its weight wi:

2. Multiply these results together:

3. Divide 1 by the sum of the weights:

4. Combine these results to find the weighted geometric mean:

So on average, each household has approximately 2.78 cellphones.

Alan Anderson, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.