Understanding Weighted Averages: Definition and Applications

The definition of a weighted average:

What is a Weighted Average?

A weighted average is the result of multiplying each number in a set by a value assigned to it (its weight), and then calculating the arithmetic mean of the resulting products. It's used when the components contributing to the average are not equally important.

For example, if a teacher states that an examination is worth 40% of the final mark, another is worth 35%, and a third is worth 25%, the weighted average would be calculated as follows:

mediaPond = (ex1 * 40 + ex2 * 35 + ex3 * 25) / 100

Basically, it's an average of a dataset that allows you to define the degree of importance for each data point's contribution to the average.

If the data are 2, 3, 5, 7, 9, 6, 8, the average is the sum of all data divided by the total number of data points. Here, each data point contributes equally to the average: (2 + 3 + 5 + 7 + 9 + 6 + 8) / 7. This becomes clearer if you consider the common mean or average as a weighted average where all data have equal importance (weight of one). This is equivalent to (2 * 1 + 3 * 1 + 5 * 1 + 7 * 1 + 9 * 1 + 6 * 1 + 8 * 1) / 7.

When determining that some data should be more important than others, you can include it in the average with a greater weight. For example, if for "x reasons" the third element should be more important than the rest, then:

(2 * 1 + 3 * 1 + 5 * 2 + 7 * 1 + 9 * 1 + 6 * 1 + 8 * 1) / 7

What you get is a weighted average. When calculating the average of pooled data, you are calculating a weighted average because each interval corresponds to a weight that determines absolute frequency (remember that the average of grouped data is (x1 * f1 + x2 * f2 + x3 * f3 + ... xn * fn) / n).

Applications of Weighted Averages

Weighted averages are commonly applied in scenarios such as:

• Calculating a student's final exam average.

Consider a scenario where each final exam is weighted four times higher than a partial exam. If a student scored 69, 75, 62, and 73 on partial exams, and 78 on the final exam, their weighted average would be:

(69 * 1 + 75 * 1 + 62 * 1 + 73 * 1 + 78 * 4) / 8