An extensive collection of numbers is represented by a single number in a simple average. But if items in the collection have different levels of importance associated with them, it might not be an accurate portrayal of an average.
In this situation, a weighted average is more accurate than a simple average. This is because each data point’s value is multiplied by the allotted weight in a weighted average before being added together and divided by the total number of data points. Thus, a weighted average can increase the accuracy of the data.
What Is a Weighted Mean?
Most frequently, a weighted average is calculated to balance the frequency of the values in a data set. For instance, a survey may have enough responses from all age groups to be deemed statistically valid, but the 18–34 age group may receive the least amount of responses compared to the proportion of the population that they make up. The survey team may factor in the data from respondents who are 18 to 34 years old to ensure that their opinions are well represented.
Values in a data set, however, may be weighted for factors other than frequency of occurrence. The mark for skill might be given more weight than the other grades, for instance, if students in a dancing class are graded on their ability, attendance, and politeness
Uses of a Weighted Mean
Weighted means are useful in a wide variety of scenarios in our daily life. Some of the common uses are as follows:
The weighted mean is used by students to determine their percentage grade in a course. In this scenario, the student must multiply the weighted average of all course assessments (such as assignments, exams, projects, etc.) by the corresponding grade that was earned in each category.
It is employed in descriptive statistical analysis, such as the computation of index numbers. For instance, the weighted average approach is used to calculate stock market indices like the Nifty or BSE Sensex. It can also be used in physics to determine an object’s centre of mass and moment of inertia.
Business owners can also benefit from comparing the average costs of items bought from various suppliers, where the weight indicates the quantity of goods purchased. It provides a clearer picture of their spending.
A customer’s decision to purchase a product or not is influenced by the product’s quality, understanding of the product, price, and service provided by the franchise. Each criterion is given a weight by the customer, who then determines the weighted average. This will enable him to choose the goods more wisely.
The interviewer evaluates a candidate’s personality, work talents, educational background, and teamwork ability to hire them for a position. Different weights (importance levels) are assigned based on the profile before the ultimate choice is decided.
What Is the Formula for Weighted Mean?
The formula to calculate the weighted mean is as follows:
Here, W denotes the weighted average, n denotes the number of terms to be averaged, wi denotes the weights applied to x values, and xi denotes the data values to be averaged.
Calculating weighted mean using formula
A trader purchases 20,000 units of a product at ₹1 each, 15,000 at ₹2 each and 10,000 at ₹3 each. We calculate the following using the units as the weight and the overall quantity of units as the total of all weights:
[1(20,000) + 2(15,000) + 3(10,000)] / (20,000 + 15,000 + 10,000) = (20,000 + 30,000 + 30,000) / (20,000 + 15,000 + 10,000) = 80,000 / 45,000 = 1.78
This equals a weighted average cost of ₹1.78 per unit.
Calculating weighted mean through excel
- Add a column to your spreadsheet that contains the weight for each data point after entering your data.
- Type =SUMPRODUCT(X:X,X:X)/SUM(X:X) and enter the values.
- Click enter to obtain your results.
How to Calculate Weighted Mean?
Suppose a firm conducts a survey of 4 shops to determine the average electricity usage in each shop per day in percentage. The first shop has one AC, which uses electricity for 40% of the day; the second shop has two ACs, which use electricity for 50% of the day; the third shop has three ACs, which use electricity for 60% of the day, and the fourth shop has four ACs, which use electricity for 70% of the day. Calculate the mean electricity usage of ACs per shop for each day.
Solution: The weighted arithmetic mean can be calculated using the steps listed below.
|
Number of ACs in the shop (xi)
|
Usage percentage in a day (wi)
|
|---|---|
|
1
|
40
|
|
2
|
50
|
|
3
|
60
|
|
4
|
70
|
Step 1: First, assign a weight to each value in the dataset.
x1 = 1, w1 = 40
x2 = 2, w2 = 50
x3 = 3, w3 = 60
x4 = 4, w4 = 70
Step 2: Calculate the numerator of the weighted mean formula.
To calculate it, multiply each sample by its weight and then add the products to obtain the final value
= 1 x 40 + 2 x 50 + 3 x 60 + 4 x 70
= 40 + 100 + 180 + 280
= 600
Step 3: Then, calculate the denominator of the weighted mean formula by adding their weights.
= 40 + 50 + 60 + 70
= 220
Step 4: At last, divide the numerator by the denominator.
= 600/220
= 2.7273
Hence, the mean electricity usage is 2.7273.
Note: An outlier in the data can readily change the weighted mean. We cannot rely on the weighted mean if our data set contains extremely high or extremely low values.
Key Takeaway
The weighted average considers the relative frequency or relevance of particular variables in a data set.
Sometimes a weighted average is more precise than a basic average.
Each data point’s value is multiplied by the allotted weight in a weighted average before being added together and divided by the total number of data points. Because of this, a weighted average can increase the accuracy of the data.
Investors in stocks use a weighted average to keep track of the cost basis of the shares they have acquired over time.
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