Weighted Arithmetic Mean

Weighted arithmetic mean is the average of all the values in a given data, with different weights given to individual values. If all the values are given the same weight, then the weighted average will be the same as the arithmetic mean. The values having higher weights have an increased influence on the average calculated from the values. All the weights given to individual values are positive or zero. But no weight can ever be negative. Weighted arithmetic mean is generally used in integral calculus, data analysis, and weighted differential calculus. Read with us to know more about this statistical value, its properties, and its applicability. 

Understanding Weighted Mean 

The weighted arithmetic mean is very similar to the arithmetic mean except that all the values do not contribute the same to the average, unlike the arithmetic mean. Some values with higher weights influence the mean value more than others in the weighted arithmetic mean calculation. 

The weighted mean helps us in finding a more accurate average by considering the weight of each value instead of only adding up individual values. Arithmetic mean is an integral statistical value used in multiple everyday purposes to make the interpretation of data easier. For instance, to find the result of a student, instead of showcasing the marks obtained in every subject in different semesters, the average of all marks is given at the end of the year. 

To understand the weighted mean in the same example, we can see how the weightage of unit tests and final exams is different in calculating the final result. Thus, to find the result, the average in all subjects gets multiplied by the weight of each unit. 

Weighted Arithmetic Mean Formula 

The mean of a data set {x1,x2,………. ,xn} , 

With the non-negative weights {w1,w2,………. ,wn} is given by, 

x=i=1nwi xii=1nwi

This formula can be expanded to: 

x=w1x1+w2x2+. . . . +wnxnw1+w2+. . . . +wn

Let us look at the difference between the arithmetic mean and the weighted mean with an example. 

Suppose there are two classes in a school, morning class, and evening class. The test grades of the 20 students in the morning class are 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98. The test grades of the 30 students in the evening class are 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99. 

Arithmetic mean = Sum of all observations / total number of observations

Thus, the arithmetic mean of the morning class will be 80 and the arithmetic mean of the evening class will be 90. The unweighted arithmetic mean of the two classes will be 85. However, this value does not consider the difference in the class strength. Thus, this value does not reflect the average marks of a student, irrespective of the class. 

To find the average grade of the students, we must sum up all the observations and divide them by the total number of children, independent of the classes. This will give us 4300 / 50 = 86. 

This can also be found by weighting the mean of each class by the number of students in the class. 

The mean will be (20 X 80) + (30 X 90) / 20 + 30 = 86

Applicability of Weighted Mean

There are multiple real-life uses of the arithmetic mean, such as: 

1. The weighted mean is used by students to find out their results at the end of a term. To find the final result, the marks in each semester or unit are multiplied by the weight of every semester to find the weighted arithmetic mean. In most classes, the unit tests have lower weightage as compared to the half-yearly exams and final exams. Thus, it becomes essential to consider the weight of every semester to find out the final result. 
2. Another usability of the weighted mean is in descriptive statistical analysis. This statistical measure is often used in the stock market indices like BSE Sensex or Nifty. Further, it is also employed in physics to drive the center mass of an object.
3. Lastly, many businessmen use the weighted mean while calculating their total purchases. They frequently purchase varied products in different quantities from merchants. To find their total purchase, they must calculate the weighted mean, with the quantity purchased acting as the weights. 

Advantages of Weighted Arithmetic Mean 

There are multiple advantages of the weighted arithmetic mean, such as:

1. Smoothen the fluctuations in readings: One of the primary benefits of this statistical value is that it smooths out the fluctuations in the stock market and other readings. Taking only the arithmetic mean might mean that the data has been affected by fluctuations, and it is not the accurate average. The weighted mean considers the time for which a price prevailed to provide furnished insights. Thus, it provides a long-run reading and a consistent valuation of the stock market purchases. 
2. Covers the uneven data: In the population study, certain segments are often under or over-represented. However, the weighted average considers the number of people being affected by an ascertained factor to give a more balanced data representation. Thus, it is an effective measure when dealing with population size and demographics readings. 
3. Simplifies the calculation: The weighted average takes the equal values as equal. There is no need to calculate the values for the same numbers individually. We can simply add up all the similar values and make the average calculation easier. The weight can be multiplied by the value to derive the weighted average. 

Thus, with the multiple applicability and advantages of the weighted average mean, it is essential to understand the concept and learn to calculate its value. The weighted average can be calculated in easy steps by finding out the mean of the individual sets and multiplying them by their weight. The resultant is then decided by the sum of the weight to provide us with the weighted average.