Weighted Average Formula

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, …… X_n:

Average = (X₁+X₂+X₃+X₄+….+X_n) / n

An extension to average, there is another famous mathematical quantity called weighted average. 

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, …… X_n. Each associated with weights w₁, w₂, w₃, w₄, …… w_n

Weighted Average = ( w₁ × X₁ + w₂ × X₂ + w₃ × X₃ + w₄ × X₄ +…..+ w_n × X_n)/(w₁ + w₂ + w₃ + w₄ +…….+ w_n)

or

W = ∑w_i × x_i/ ∑w_i

or

W = Sum of (Product of weights with their respective quantities)/ Sum of all the weights

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 = ∑w_i = 5 + 3 + 5 + 6 = 19

∑w_i  x_i = 4×5 + 6×3 + 2×5 + 3×6

= 20 + 18 + 10 + 18 

= 66

W = ∑w_i×x_i/ ∑w_i

= 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 = ∑w_i = 40 + 30 = 70

∑w_i×x_i

= 70×40 + 50×30

= 2800 + 1500

= 4300

W = ∑w_i×x_i/ ∑w_i

= 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 = ∑w_i = 6 + 2 + 3 + 9 = 20

∑w_i×x_i = 80×6 + 45×2 + 60×3 + 20×9

= 480 + 90 + 180 + 180

= 930

W = ∑w_i×x_i/ ∑w_i

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 = ∑w_i = 5 + 3 + 6 + 3 + 7 = 24

∑w_i×x_i = 3×5 + 6×3 + 8×6 + 4×3 + 2×7

= 15 + 18 + 48 + 12 + 14

= 107

W = ∑w_i×x_i/ ∑w_i

= 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 = ∑w_i = 2 + 3 + 1 + 2 + 4 + 2 = 14

∑w_i×x_i = 20×2 + 30×3 + 40×1 + 50×2 + 60×4 + 70×2

= 40 + 90 + 40 + 100 + 240 + 140

= 650

W = ∑w_i×x_i/ ∑w_i

= 650/14

= 325/7