3.6:
Weighted Mean
Consider a data set where some values are more important than others; in other words, they carry more weight.
To calculate the mean for such data, each value is multiplied by its weight. The resulting products are added and then divided by the sum of weights. This is called the weighted mean.
For example, a student takes several tests in a year, each weighted differently. To determine the weighted mean of all the tests, multiply the individual test scores with the corresponding weights and add these products. Then, divide this final value by the sum of all the weights.
As one can see, the students get better mean scores by doing well in tests with higher weights. That means, the data values with a higher weight contribute more to the weighted mean.
If the weights of all the data values are the same, then the weighted mean is equal to the arithmetic mean.
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.