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:
Compute the value of each Xi with an exponent equal to its weight wi:
Multiply these results together:
Divide 1 by the sum of the weights:
Combine these results to find the weighted geometric mean:
So on average, each household has approximately 2.78 cellphones.