Sometimes a data set contains a large number of repeated values. In these situations, 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 compute the arithmetic mean as a weighted average.

The formula for computing a weighted arithmetic mean for a sample or a population is

Here, *w** _{i}* represents the

*weight*associated with element

*X*

*; this weight equals the number of times that the element appears in the data set.*

_{i}The *numerator* (the top half of the formula) tells you to multiply each element in the data set by its weight and then add the results together, as shown here:

The *denominator* (the bottom half of the formula) tells you to add the weights together:

You find the weighted arithmetic mean by dividing the numerator by the denominator.

As an example, suppose that a marketing firm conducts a survey of 1,000 households to determine the average number of TVs each household owns. The data show a large number of households with two or three TVs and a smaller number with one or four. Every household in the sample has at least one TV and no household has more than four. Here's the sample data for the survey:

Number of TVs per Household | Number of Households |
---|---|

1 | 73 |

2 | 378 |

3 | 459 |

4 | 90 |

Because many of the values in this data set are repeated multiple times, you can easily compute the sample mean as a weighted mean. Doing so is quicker than summing each value in the data set and dividing by the sample size.

Follow these steps to calculate the weighted arithmetic mean:

Assign a weight to each value in the data set:

*X*_{1}= 1,*w*_{1}= 73*X*_{2}= 2,*w*_{2}= 378*X*_{3}= 3,*w*_{3}= 459*X*_{4}= 4,*w*_{4}= 90Compute the numerator of the weighted mean formula.

Multiply each sample by its weight and then add the products together:

Compute the denominator of the weighted mean formula by adding the weights together:

Divide the numerator by the denominator:

The mean number of TVs per household in this sample is 2.566.