3.6: Weighted Mean
While taking the arithmetic, geometric, or harmonic mean of a sample data set, equal importance is assigned to all the data points. However, all the values may not always be equally important in some data sets. An intrinsic bias might make it more important to give more weightage to specific values over others.
For example, consider the number of goals scored in the matches of a tournament. While computing the average number of goals scored in the tournament, it may be more important to consider the games played in its knockout stage. The goals from the knockout stage may carry more weight than the other goals. Once a numerical estimate is assigned to this idea, the average number of goals in the tournament is calculated. Such means are called weighted means. They help us assign an intrinsic value to different elements of a data set.
Sometimes, the probability of occurrence of each element can play the role of weights. For example, if biased dice are thrown at random a few times, some numbered sides may appear more frequently than the others. The weighted mean of the numbers accounts for this bias.