We explain what the weighted average is in statistics and mathematics, examples and the steps to obtain it.
What is the weighted average?
In math Y statistics, the weighted average or weighted mean is the measure of central tendency obtained from a set of data whose relevance or importance within the group is relative to the others.
That is, when we have a series of data that do not have the same relevance (that is, they do not have the same weighing) inside of the set, so it is not appropriate to simply obtain an arithmetic mean.
Thus, to obtain a weighted average we must multiply each data by its weight (or weight) and then add them (this is called a weighted sum), to finally divide the figure obtained by the sum of the weights or weights. This is much easier to observe with an example:
Suppose that to pass his math course, a student must take three partial exams and one final exam, each of which corresponds to a different score in the final grade for the course. Thus, each of the partial exams is equivalent to 2 points and the final exam, on the other hand, corresponds to 4 points, for a total of 10 possible points in the final grade of the course (2 + 2 + 2 + 4 = 10).
So, at the end of the semester, the student has obtained the following grades in his midterm exams: 6, 5, 3. The subject, obviously, is not given to him. But on the final exam, for which he studied as hard as he could, he got a very decent 7. What will his weighted average be?
Let's first obtain the weighted sum of his exams: (6 x 2) + (5 x 2) + (3 x 2) + (7 x 4) = 12 + 10 + 6 + 28 = 56. This figure must then be divided by the sum of all the weightings, that is, as we already knew, 10. Thus, the student's weighted average will be 56 / 10, which is equivalent to 5.6 points. He passed right on the edge!
Note that the simple arithmetic mean of these grades (6 + 5 + 3 + 7 divided by 4) would result in 5.25. This figure would be inaccurate because it assigns the same value to all exams, and the final exam obviously has greater relevance because the student must respond to the total content of the subject.
Other Weighted Average Examples
Here are a couple more examples to understand how the weighted average is calculated:
- An investor buys shares of different companies that represent percentages different from the total shareholders of each one: 100 shares in Tecnocorp representing 20% of the total; 50 shares in Medlab S.A. representing 5% of the total, and 500 shares in Politruck Inc. representing 50% of the total. What is the weighted average amount invested?
Again, to solve this we must obtain a addition weighted first: (100 x 20) + (50 x 5) + (500 x 50) = 2,000 + 250 + 25,000 = 27,250, and then divide the figure by the sum of the weights (20 + 5 + 50 = 75 ). Thus, the weighted average of the shares purchased will be 363.33.
- A miner obtains gold fragments of different purity grades: three fragments of 50% purity, two of 60% and one of only 90%. What is the weighted average of the obtained?
Weighted sum: (3 x 50) + (2 x 60) + (1 x 90) = 150 + 120 + 90 = 360, between the sum of the purity percentages: 50 + 60 + 90 = 200. The weighted average of the gold obtained will then be 1.8%.