The Weighted average or weighted arithmetic mean, is a measure of central tendency in which, at each value xi which can take a variable X, it is assigned a weight pi. As a result, denoting the weighted mean by xp, you have:
With summation notation, the formula for the weighted average is:
Where N represents the number of values that are chosen from the variable X.
The pi, who is also called weighting factor, it is a measure of the importance that the researcher assigns to each value. This factor is arbitrary and always positive.
In this, the weighted mean differs from the simple arithmetic mean, because in this, each of the x valuesn has equal significance. However, in many applications, the researcher may consider that some values are more important than others and will assign a weight to them according to their criteria..
Here is the best known example: suppose that a student takes N assessments in a subject and they all have the same weight in the final grade. In this case, to calculate the final grade it will be enough to take a simple average, that is, add all the qualifications and divide the result by N.
But if each activity has a different weight, because some evaluate more important or more complex content, then it will be necessary to multiply each evaluation by its respective weight, and then add the results to obtain the final grade. We will see how to carry out this procedure in the solved exercises section.
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The example of the ratings described above is one of the most typical in terms of the application of the weighted average. Another very important application in economics is the consumer price index or consumer's price index IPC, also called family basket and that serves as an evaluator of inflation in an economy.
In its preparation, a series of items such as food and non-alcoholic beverages, clothing and footwear, medicines, transport, communications, education, leisure and other goods and services are taken into account..
The experts assign a weighting factor to each item, according to its importance in people's lives. Prices are collected during a set period of time, and with all the information the CPI for said period is calculated, which can be monthly, bimonthly, semi-annually or annually, for example..
In physics, the weighted average has an important application, which is to calculate the center of mass of a particle system. This concept is very useful when working with an extended body, in which its geometry must be taken into account..
The center of mass is defined as the point at which all the mass of an extended object is concentrated. On this point, forces such as weight, for example, can be applied and thus their translational and rotational movements can be explained, using the same techniques that were used when all objects were assumed to be particles..
For simplicity, we begin by assuming that the extended body is composed of a quantity N of particles, each with mass m and its own location in space: the coordinate point (xi, Yi, zi).
Be xCM the coordinate x from the center of mass CM, then:
M represents the total mass of the system. We proceed in the same way to find the coordinates andCM and ZCM:
The weighting factor in this case is the mass of each of the particles that make up the extended object.
When the number of particles is very large, it is a continuous object. In this case N → ∞ and the summation is replaced by a definite integral, whose limits are given by the size of the object.
It is important to note the fact that there is not necessarily mass at the location of the center of mass. For example in a donut or donut, the center of mass more or less coincides with the geometric center of the donut.
The location of the center of mass also does not depend on the reference system that is used to establish the positions of the particles, since it is a property that depends on the configuration of the object itself and not on how it is seen from different reference frames..
In many cases, teachers assign different weights or percentages to each assessment activity in their chair. Thus, for example, the tasks have a percentage, the short exams another different and the end-of-course exam one probably much higher..
Suppose that in a certain subject, the assessment activities and their respective weights are the following:
-Homework: 20%
-Short exams: 25%
-Lab Reports: 25%
-Final exam: 30%
a) How does the teacher calculate the final grade for this subject for each student?
b) Suppose that the grades of a particular student are, on a scale of 1 to 5, the following:
-Tasks: 5.0 points
-Short exams: 4.7 points
-Laboratory Reports: 4.2 points
-Final exam: 3.5 points
Find the final grade of the student in this subject.
a) Each assessment has a different weight, which the teacher has assigned according to their complexity and at their own discretion. In this way, the final grade is calculated directly as:
Definitive = (Homework x20% + Short exams x25% + Reports x25% + Final exam x30%) / 100
b) Definitive = (5.0 x 0.2) + (4.7 x 0.25) + (4.2 x 0.25) + (3.5 x 0.3) points = 4.275 points ≈ 4.3 points
The owners of a clothing store bought jeans from three different suppliers.
The first sold 12 units at a price of € 15 each, the second 20 units at € 12.80 each and a third bought a batch of 80 units at € 11.50.
What is the average price the shop owners have paid for each cowboy?
xp = (12 x 15 + 20 x 12.80 +80 x 11.50) / (12 + 20 + 80) € = 12.11 €
The value of each jean is € 12.11, although some cost a little more and others a little less. It would have been exactly the same if the store owners had bought the 112 jeans from a single vendor who sold them for € 12.11 a piece.
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