Weighted Mean Formula

Mean is also called average in Mathematics which denotes the sum of all given quantities divided by the number of quantities. The arithmetic mean is important in statistics. For example, Let’s say there are only two quantities involved, the arithmetic mean is obtained simply by adding the quantities and dividing by 2.

Weighted mean is a statistical measure used to determine the average of a set of numbers, where each number is multiplied by a predetermined weight before calculating the average.

If there are n quantities, q₁, q₂, q₃, q₄, …… q_n, then mean of quantities is denoted by  and is calculated by,

= (q₁+ q₂+ q₃+ q₄+….+ q_n) / n

Weighted Mean for quantities is different from the mean as, in the calculation of weighted mean, each quantity in the calculation of weighted means is assigned a weight w_x. This weight is different for different quantities and more specifically, this weight can be some kind of priority or entity associated with quantities.

Suppose the Given quantities are q₁, q₂, q₃, q₄, …… q_n

And weights associated with them are w₁, w₂, w₃, w₄, …… w_n

Then Weighted Mean is given by

Weighted Mean = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n )/ (w₁+ w₂+ w₃ +…….w_n)

To calculate the weighted mean follow the steps added below:

Step 1: Multiply each number by its corresponding weight.

Step 2: Add up all the products.

Step 3: Divide the sum of the products by the sum of the weights.

This explained using an example added below:

Example: In three exams in a course, each exam has a different weight in determining your final grade. Here are the scores and weights:

• Exam 1: Score = 80, Weight = 30%
• Exam 2: Score = 90, Weight = 40%
• Exam 3: Score = 85, Weight = 30%

Find weighted mean:

Solution:

To calculate the weighted mean (your overall grade), weuse the formula:

Weighted Mean = {(80×.3) + (90×.4) + (85×.3)}/(0.3 + 0.4 + 0.3)

= {24 + 36 + 25.5}/1 = 85.5

So, the weighted mean of your exam scores is 85.5.

Weighted mean and the arithmetic mean are explained using a table for better visualization.

Let’s consider the following data set:

Arithmetic Mean Calculation:

{Arithmetic Mean} = {Sum of Values}\{Number of Values}}

Sum of Values = (10×1) + (20×1) +(30×1) = 10+20+30 = 60

Number of Values=3

Arithmetic Mean = 60/3 = 20

Weighted Mean Calculation:

Weighted Mean = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n )/ (w₁+ w₂+ w₃ +…….w_n)

Sum of (Values × Weights) = (10×2) + (20×3) + (30×5) = 20 + 60 + 150 = 230

Sum of Weights = 2+3+5 = 10

Weighted Mean = 230/10 = 23

So, the arithmetic mean for this data set is 20, while the weighted mean is 23. This shows how the weights affect the outcome, pulling the mean higher due to the emphasis placed on the larger values.

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Example 1: Given quantities 10, 20, 30, and 40 are each associated with a weight of 2, 3, 4, and 5. Find the weighted mean of the quantities.

Solution:

Weighted Mean is given by Formula = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

So,

Weighted Mean = (10×2+ 20×3 + 30×4 + 40×5)/ (2 + 3+ 4+ 5)

= (20 + 60 + 120 + 200)/ 14 = 400/ 14

= 28.57

Example 2: Given quantities 50, 25, 36, and 41 are each associated with a weight of 2.5, 8, 6, and 5. Find the weighted mean of the quantities.

Solution:

Weighted Mean is given by Formula = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

So,  

Weighted Mean  = (50×2.5 + 25×8 + 36×8 + 41×5)/ (2.5 + 8 + 6 + 5)

= (125 + 200 + 288 + 205)/ 21.5

= 818/ 21.5

= 38.046

Example 3: Given quantities 5, 15, 20, 22, and 30 are each given a priority entity weight 1, 2, 3, 4, 5. Find the weighted mean of the quantities.

Solution:

Weighted Mean is given by Formula = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

So,

Weighted Mean  = (5×1 + 15×2 + 20×3 + 22×4 + 30×5)/ (1 + 2 + 3 + 4 + 5)

 = (5 + 30 + 60 + 88 + 150)/ 15

= 333/ 15 = 22.2

Example 4: Given quantities 3,4,5 is each associated with a weight 2,2,3. Find the weighted mean of the quantities.

Solution:

Weighted Mean is given by Formula = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

So,

Weighted Mean = (3×2 + 4×2 + 5×3)/ (2 + 2 +3) 

= (6 + 8 + 15)/ 7

= 29/ 7 = 4.142

Example 5: Given quantities 64, 32, 81, 49, 56, 65 is each given a priority entity weight 2, 1, 3, 4, 3, 5. Find the weighted mean of the quantities.

Solution:

Weighted Mean is given by Formula = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

So,

Weighted Mean = (64×2 + 32×1 + 81×3 + 49×4 + 56×3 + 65×5)/ (2 + 1 + 3 + 4 + 3 + 5)

= (128 + 32 + 243 + 196 + 168 + 325)/ 18

= 1092/ 18 = 60.66

What is the Weighted Mean Formula?

Weighted mean formula is:

Weighted Mean = (w_1×q₁ + w_2×q₂ + w_3×q₃……………….+ w_n×q_n)/ (w₁+ w₂+ w₃ +…….w_n)

How do you Calculate the Weighted Mean?

Weighted mean is calculated by multiplying each data point value by its weight, then finding its sum and then dividing the same by total sum of weight.

What is an Example of a Weighted Mean?

An example of calculating a weighted mean for students’ grades in a course where different assignments have different weights.