An average in which each quantity to be averaged is assigned a weight. These weights determine the relative importance of each quantity on the average. Weights are the equivalent of having that many like items with the same value involved in the average.

One application for a weighted average is if you have to average a large number of items which have relatively few different values, and you are given a distribution of the values, which might look like the following:

value  occurrences
 10        2
  8        2
  5        1
  4       10
  3        8
  2        7
  1       68
  0        2

These are the values of letter tiles in Scrabble. To average these values, do a weighted average using the number of occurrences of each value as the weight.

To calculate a weighted average:

  1. Multiply each value by its weight.
  2. Add up the products of value times weight to get the total value.
  3. Add the weights themselves to get the total weight.
  4. Divide the total value by the total weight.
For the data above, you'd multiply and get products of 20, 16, 5, 40, 24, 14, 68, and 0, and add these to get a total value of 187. Then you'd add the weights to get 100, and divide to get the average value of a Scrabble tile, 1.87 points.

Note that the normal average, or arithmetic mean, is equivalent to a weighted average in which all the weights are 1.

In the example of the scrabble tiles, given by /dev/joe above, the weights used (occurences) led us to getting the exact arithmetic mean of values all the scrabble tiles. This is not necessarily true for all weighted average.

The weights are more generally related to some measures of importance you gave the individual values in order to make up the desired weighted average. In such cases the weighted average would not be at all directly related to the normal average or arithmetic mean.

For example, the weights could be related to the age of the value, where the recent values are considered more important, such as in Weighted Average with denominator 7.

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