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The weighted mean is a type of average where each value in a dataset is multiplied by a specific weight before the final calculation. This method is useful when certain values in the dataset contribute more significantly to the overall average than others.
The weighted mean is similar to the arithmetic mean. In arithmetic mean, each data point contributes equally to the final average but whereas in weighted mean few data points contribute more to the resultant average. Here each data point is associated with some weight. Depending on the weights of observations the contribution in the final average varies.
Let’s know more about Weighted Mean definition, formula and solved examples in detail below.
Weighted mean is a statistical method that is calculated by multiplying the weight by the quantitative outcome associated with it and then adding all the products together. This resultant is divided by the sum of all weights associated with the observations giving a weighted mean. If all the weights of observations are the same then the arithmetic mean is equal to the weighted mean.
Let’s consider the data points x1,x2,x3,…,xn which are associated with weights w1,w2,w3,…,wn then the weighted mean can be calculated by the formula
Weighted Mean = ∑in=1 xi.wi/∑in=1 wi
=(x1w1+x2w2+x3w3+…+xnwn)/(w1+w2+w3+…+wn)
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Steps to calculate Weighted MeanFollow these steps to calculate the Weighted Mean.
- Tabulate the given data that helps for easy calculations.
- Find wi×xi by multiplying each number with the associated weight.
- Calculate the summation of all products calculated in step-2 to give ∑wi×xi.
- Find the sum of all weights i.e., ∑wi.
- Divide the total value obtained in step-3 ∑wi×xi with the value obtained in step-4 ∑wi to give the final result weighted mean.
Differences between Weighted Mean and Arithmetic MeanBelow is the Weighted Mean and Arithmetic Mean difference mentioned in below table.
| Aspect | Weighted Mean | Arithmetic Mean |
|---|
| Definition | Average where each value is multiplied by a specific weight | Simple average of a set of values | Formula
|
[Tex]\frac{\sum{x_{i}}}{n}[/Tex]
| \frac{{\sum{(x_{i}-v_{i})}}}{v_{i}}
| | Values Used | Values are multiplied by weights | All values are treated equally | | Equal Importance | Accounts for different levels of importance | Assumes all values have equal importance | | Calculation Steps | 1. Multiply each value by its weight
2. Sum the products \newline 3. Sum the weights
4. Divide the sum of products by the sum of weights | 1. Sum all values
2. Divide by number of values | | Use Cases | Finance, economics, research | Average grades, temperatures | | Impact of Values | Higher weights have a greater impact on the mean | Every value equally affects the mean | | Application Example | Average price of stocks with different shares | Average score of a class |
Sample Questions on Weighted MeanQuestion 1: Find the weighted mean for the given data
Value
| Weight
| 10
| 4
| 5
| 3
| 20
| 2
| 15
| 6
| 8
| 10
|
Solution:
Given values associated with weights, calculate ∑wixi and ∑wi to find weighted mean.
xi
| wi
| xiwi
| 10
| 4
| 40
| 5
| 3
| 15
| 20
| 2
| 40
| 15
| 6
| 90
| 8
| 10
| 80
|
| ∑wi=25
| ∑wixi=265
|
Weighted mean = ∑wixi/∑wi
= 265/25
= 10.6
Weighted mean for the given data is 10.6
Question 2: Find the weighted mean for the given data
Value
| Weight
| 5
| 5
| 15
| 2
| 25
| 1
|
Solution:
Given values associated with weights, calculate ∑wixi and ∑wi to find weighted mean.
xi
| wi
| xiwi
| 5
| 5
| 25
| 15
| 2
| 30
| 25
| 1
| 25
|
| ∑wi=8
| ∑wixi=80
|
Weighted mean = ∑wixi/∑wi
= 80/8
= 10
Weighted mean for the given data is 10
Question 3: Find the weighted mean for the given data
Value
| Weights
| 2
| 4
| 4
| 3
| 6
| 2
| 8
| 1
|
Solution:
Given values associated with weights, calculate ∑wixi and ∑wi to find weighted mean.
xi
| wi
| xiwi
| 2
| 4
| 8
| 4
| 3
| 12
| 6
| 5
| 30
| 8
| 1
| 8
| | | ∑wi=13
| ∑wixi=58
|
Weighted mean = ∑wixi/∑wi
= 58/13
= 4.46
Weighted mean for the given data is 4.46
Question 4: Find the weighted mean for the given data
Value
| Weight
| 80
| 0.2
| 90
| 0.4
| 70
| 0.5
|
Solution:
Given values associated with weights, calculate ∑wixi and ∑wi to find weighted mean.
xi
| wi
| xiwi
| 80
| 0.2
| 16
| 90
| 0.4
| 36
| 70
| 0.5
| 35
|
| ∑wi=1.1
| ∑wixi=87
|
Weighted mean = ∑wixi/∑wi
= 87/1.1
= 79.09
Weighted mean for the given data is 79.09
Question 5: Find the weighted mean for the given data
Values
| Weights
| 72
| 2
| 66
| 1
| 76
| 1
| 54
| 4
| 62
| 3
|
Solution:
Given values associated with weights, calculate ∑wixi and ∑wi to find weighted mean.
xi
| wi
| xiwi
| 72
| 2
| 144
| 66
| 1
| 66
| 76
| 1
| 76
| 54
| 4
| 216
| 62
| 3
| 186
|
| ∑wi=11
| ∑wixi=688
|
Weighted mean = ∑wixi/∑wi
= 688/11
= 62.54
Weighted mean for the given data is 62.54
How to calculate the Weighted Mean – FAQsWhat is Weighted Mean?The weighted mean is a type of average where each value in a dataset is multiplied by a specific weight before the final calculation. This method is useful when certain values in the dataset contribute more significantly to the overall average than others.
How to calculate Weighted Mean?Calculating the weighted mean involves several steps to ensure each value in your dataset is properly accounted for based on its importance or frequency.
- List the values and their corresponding weights: Make sure you have each value and its associated weight clearly listed.
- Multiply each value by its weight: For each value, calculate the product of the value and its weight. This gives you the weighted values.
- Sum the weighted values: Add up all the weighted values from the previous step to get the total weighted sum.
- Sum the weights: Add up all the weights used in your calculations. This gives you the total weight.
- Divide the total weighted sum by the total weight: The final step is to divide the total weighted sum by the total weight to get the weighted mean.
The formula for weighted mean is [Tex]\frac{\sum{x_{i}}}{n}[/Tex]
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