Statistical data analysis would greatly benefit from accurately representing a collection of information. The weighted average formula is one such useful tool. It does not simply treat all points of data like an ordinary average but gives distinct values with different weights.
Such a method finds application in cases where specific numbers have more significance than others in a dataset. Hence, for instance, it could be applicable in examples like financial portfolios or school grades among others; thus if you want your decisions based on better knowledge and exactness mastering the weighted average formula is fundamental.
What is the Weighted Average?
The weighted average is a type of average where each value in a data set is multiplied by a predetermined weight before the final calculation. These weights are taken as the measures of relative importance or frequency with which each figure occurred within that particular observation series.
Whereas simple average considers all amounts equally, this type of measure reflects their significance and thus it may provide a better picture than others in some cases.
Formula for Weighted Average
Weighted Average=\frac{\Sigma(w_i\cdot x_i)}{\Sigma w_i}
where:
- w_i represents the weight assigned to each value x_i.
- x_i represents the individual values in the data set.
- \Sigma denotes the sum over all values.
Example
To illustrate the calculation of a weighted average, consider a student's final grade composed of homework, midterms, and finals, each with different weights. The table below shows the scores and corresponding weights for each component:
Component | Score (x_i) | Weight (w_i) | Weighted Score (w_i\cdot x_i) |
|---|
Homework | 85 | 0.20 | 17 |
|---|
Midterm | 90 | 0.30 | 27 |
|---|
Final | 80 | 0.50 | 40 |
|---|
To calculate the weighted average:
- Sum of Weighted Scores (\Sigma(w_i\cdot x_i)): 17+27+40=84
- Sum of Weights \Sigma w_i: 0.20+0.30+0.50=1
- Weighted Average: \frac{84}{1}=84
Therefore, the student's final weighted average grade is 84.
This tabular representation clearly shows how each component's score is multiplied by its respective weight and how the sum of these weighted scores is used to compute the weighted average.
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 pera 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.
Conclusion
In statistics it is used to show values regarding things or describe weights based on their importance. Using the different weights attached to data, we can view a clearer picture especially in cases where some things matter more than others do.
The application of a formula for weighted average enables people to make decisions based on total appreciation and accurate data understanding ranging from academic grades calculations to financial portfolio valuation. Those who know how to use the weighted average effectively can improve their decision-making abilities and achieve optimal results in diverse disciplines as well as insights into them. Through the weighted average implementation one can get a more thorough and authentic analysis at a time when everything in question is accounted for significantly within its respective data set.
Solved Questions
Question 1: The different quantities and the associated weights with them are as given below. Find their weighted average.
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.
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:
Practice Problems - Weighted Average Formula
1. A student's final grade is based on three components: assignments (worth 25%), midterm exam (worth 35%), and final exam (worth 40%). The student scored 80 on assignments, 75 on the midterm, and 90 on the final exam. Calculate the weighted average of the student's scores.
2. A company evaluates its employee performance based on three criteria: productivity (40%), teamwork (30%), and punctuality (30%). If an employee scores 85 in productivity, 90 in teamwork, and 80 in punctuality, what is their weighted average performance score?
3. In a portfolio, an investor holds three stocks with the following values and weights: Stock A (value: $50, weight: 40%), Stock B (value: $30, weight: 35%), and Stock C (value: $20, weight: 25%). Calculate the weighted average value of the portfolio.
4. A school calculates GPA based on grades in four subjects with the following weights: Mathematics (weight: 30%), Science (weight: 25%), English (weight: 25%), and History (weight: 20%). If a student has grades of 88 in Mathematics, 92 in Science, 85 in English, and 90 in History, what is their GPA?
5. A restaurant rates its food items based on taste (weight: 40%), presentation (weight: 30%), and value for money (weight: 30%). If a dish scores 9 for taste, 8 for presentation, and 7 for value for money, what is the weighted average rating for the dish?
6. An athlete's overall performance score is based on three events: sprint (weight: 50%), long jump (weight: 30%), and high jump (weight: 20%). The athlete scores 85 in sprint, 90 in long jump, and 80 in high jump. Calculate the weighted average performance score.
7. A product's overall rating is determined by three factors: durability (weight: 40%), design (weight: 35%), and cost (weight: 25%). If the product scores 7 in durability, 8 in design, and 6 in cost, what is the weighted average rating?
8. A university's admission committee considers three criteria for student selection: academic performance (weight: 60%), entrance test (weight: 30%), and extracurricular activities (weight: 10%). If a student has an academic score of 90, an entrance test score of 85, and an extracurricular score of 80, calculate their weighted average admission score.
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