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 3^{4}; 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

*X*with an exponent equal to its weight_{i}*w*:_{i}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.