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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.