Weighted Average: How It Is Calculated, Examples And Exercises

The weighted average or weighted arithmetic mean is a measure of central tendency in which, to each value x i that a variable X can take, a weight p i is assigned . As a result, denoting the weighted mean by x p , we have:

Using summation notation, the formula for the weighted mean is:

Where N represents the number of values ​​that are chosen from the variable X.

The p i, which is also called the weighting factor,  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 n values 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 at their discretion.

Here is the most popular example: suppose 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.

Article index

  • one

    Examples

    • 1.1

      The center of mass of a system of particles

  • two

    Solved exercises

    • 2.1

      – Exercise 1

    • 2.2

      – Exercise 2

  • 3

    References

Examples

Figure 1. The weighted average is applied when calculating the consumer price index, an indicator of inflation. Source: PxHere.

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 CPI consumer price index, also called the family basket and which 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.

The center of mass of a system of particles

In physics, the weighted average has an important application, which is calculating 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 number N of particles, each with mass m and its own location in space: the point of coordinates (x i , y i , z i ).

Let x CM be the x coordinate of the center of mass CM, then:

M represents the total mass of the system. We proceed in the same way to find the coordinates y CM and z CM :

The weighting factor in this case is the mass of each of the particles that make up the extended object.

Important characteristics of the center of mass

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.

Importantly, there is not necessarily mass at the location of the center of mass. For example in a donut or donut, the center of mass roughly coincides with the geometric center of the donut.

Figure 2. The center of mass of a donut, a fairly symmetrical object, is in the hole. Source: Pixabay.

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.

Solved exercises

– Exercise 1

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.

Figure 3. In their assessment plans, teachers often assign different weights to assessments. Source: Grade book by David Mulder via Flickr.

Suppose that in a certain subject, the assessment activities and their respective weights are the following:

-Homework: 20%

-Short exams: 25%

-Laboratory 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.

 Solution

a) Each evaluation has a different weight, which the teacher has assigned according to their complexity and his own criteria. 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

– Exercise 2

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 store owners have paid for each cowboy?

Solution

x p = (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.

References

  1. Arvelo, A. Measures of Central Tendency. Recovered from: franarvelo.wordpress.com
  2. Mendenhall, W. 1981. Statistics for Management and Economics. 3rd. edition. Grupo Editorial Iberoamérica.
  3. Moore, D. 2005. Applied Basic Statistics. 2nd. Edition.
  4. Triola, M. 2012. Elementary Statistics. 11th. Ed. Pearson Education.
  5. Wikipedia. Weighted average. Recovered from: en.wikipedia.org

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