Average in mathematics holds major significance. Ancient mathematics describes average as the quantity same as a single quantity describing all the available given quantities.
For given quantities X1, X2, X3, X4, …… Xn:
Average = (X1+X2+X3+X4+….+Xn) / n
An extension to average, there is another famous mathematical quantity called weighted average.
What is the Weighted Average?
- In weighted average, each quantity is assigned a weight, which is multiplied by the quantity to calculate the average.
- The weight assigned to these quantities can be a priority number or any other entity which describes some importance to the given quantities.
- All the quantities while calculating the average were equal, but here in calculating the weighted average, the quantities are assigned different weights.
For given quantities X1, X2, X3, X4, …… Xn. Each associated with weights w1, w2, w3, w4, …… wn
Weighted Mean Formula
Weighted Average = ( w1 × X1 + w2 × X2 + w3 × X3 + w4 × X4 +…..+ wn × Xn)/(w1 + w2 + w3 + w4 +…….+ wn)
or
W = ∑wi × xi/ ∑wi
or
W = Sum of (Product of weights with their respective quantities)/ Sum of all the weights
Formula to Calculate Weighted Average in Excel
You can easily compute a weighted average in Excel by following these steps:
Utilize the “SUMPRODUCT” and “SUM” functions to determine the weighted average of a given dataset.
For instance, let’s consider the price of 10 apples at 20 rupees and another fruit at 40 rupees per 10 fruits. To calculate the weighted average, apply the formula “=SUMPRODUCT(A2:A3, B2:B3)/SUM(B2:B3)” where A2 and A3 represent the frequency and quantity of the items respectively, and B2 and B3 denote the respective values for the other variety of fruits in the dataset.
Solved Questions
Question 1: The different quantities and the associated weights with them are as given below. Find their weighted average.
| Quantity | WEIGHT |
| 4 | 5 |
| 6 | 3 |
| 2 | 5 |
| 3 | 6 |
Answer:
Sum of weights = ∑wi = 5 + 3 + 5 + 6 = 19
∑wi xi = 4×5 + 6×3 + 2×5 + 3×6
= 20 + 18 + 10 + 18
= 66
W = ∑wi×xi/ ∑wi
= 66 /19
Question 2: The different quantities and the associated weights with them are as given below. Find their weighted average.
| DATA VALUE | WEIGHT |
| 70 | 40 |
| 50 | 30 |
Answer:
Sum of weights = ∑wi = 40 + 30 = 70
∑wi×xi
= 70×40 + 50×30
= 2800 + 1500
= 4300
W = ∑wi×xi/ ∑wi
= 4300/70
= 430/7
Question 3: The data values and the associated weights with them are as follows:
| DATA VALUE | WEIGHT |
| 80 | 6 |
| 45 | 2 |
| 60 | 3 |
| 20 | 9 |
Answer:
Sum of weights = ∑wi = 6 + 2 + 3 + 9 = 20
∑wi×xi = 80×6 + 45×2 + 60×3 + 20×9
= 480 + 90 + 180 + 180
= 930
W = ∑wi×xi/ ∑wi
W = 930/20
= 93/2
Question 4: The different quantities and the associated weights with them are as follows:
| Quantities | Weights |
| 3 | 5 |
| 6 | 3 |
| 8 | 6 |
| 4 | 3 |
| 2 | 7 |
Answer:
Sum of weights = ∑wi = 5 + 3 + 6 + 3 + 7 = 24
∑wi×xi = 3×5 + 6×3 + 8×6 + 4×3 + 2×7
= 15 + 18 + 48 + 12 + 14
= 107
W = ∑wi×xi/ ∑wi
= 107/24
Question 5: The quantities and the associated weights with them are as follows:
| Quantity Value | Weight |
| 20 | 2 |
| 30 | 3 |
| 40 | 1 |
| 50 | 2 |
| 60 | 4 |
| 70 | 2 |
Answer:
Sum of weights = ∑wi = 2 + 3 + 1 + 2 + 4 + 2 = 14
∑wi×xi = 20×2 + 30×3 + 40×1 + 50×2 + 60×4 + 70×2
= 40 + 90 + 40 + 100 + 240 + 140
= 650
W = ∑wi×xi/ ∑wi
= 650/14
= 325/7
Related Topics:
- Equilateral Triangle Formula
- Percentile Formula
- Selling Price Formula
- Diameter Formula
- Volume Formulas
- Triangle Formula
- Cos Theta Formula
- Sin Cos Formula
- Pyramid Formula
Weighted Average Formula – FAQs
What is a weighted average and why is it used?
A weighted average is a calculation that assigns different weights to different elements in the dataset, reflecting their varying levels of importance. This method is commonly used in scenarios like calculating academic grades where different assignments or exams have different significance, or in finance to determine the average cost of shares purchased at different prices.
How do you calculate a weighted average?
To calculate a weighted average, multiply each number in your dataset by its corresponding weight, sum these products, and then divide by the sum of the weights. This method helps to reflect the significance or frequency of each data point more accurately than a simple average.
Can you give an example of where a weighted average might be applied?
A weighted average is particularly useful in education and finance. For instance, teachers may calculate a student’s final grade by weighting different types of assessments (tests, quizzes, projects) based on their importance to the course. Similarly, investors might use a weighted average to determine the average price of shares bought at different times and prices.
What are the steps involved in the calculation of a weighted mean?
The steps to calculate a weighted mean are as follows: List all numbers and their corresponding weights in a table for clarity, multiply each number by its weight, sum these products, and then divide this sum by the total sum of the weights. This process gives you the weighted mean of the dataset (vedantu.com).
How does a weighted mean differ from a simple average?
Unlike a simple average where each data point contributes equally to the final result, a weighted mean assigns more importance to certain data points through the use of weights. This makes the weighted mean a more flexible and representative measure, especially when data points are not equally relevant.