Weighted arithmetic mean assigns different weights or levels of importance to values before calculating the average. It is calculated by multiplying each value by its weight, summing these products, and dividing by the total of the weights. Combined arithmetic mean calculates the average of two or more groups by taking the weighted average of the means of each group, where the weights are the number of items in each group. Weighted and combined arithmetic means are useful when not all values have equal importance or when calculating averages of related subgroups.

1. WEIGHTED ARITHMETIC MEAN &
COMBINED ARITHMETIC MEAN
Dr. Mamatha S Upadhya

2. WEIGHTED ARITHMETIC MEAN
Usually in computing Arithmetic Mean, equal importance is given to all the observations of
the data. However there are cases where all the items are not of equal importance. In other
words some items of a series are more important as compared to the other items in the same
series.
Def: Weighted mean is the mean of a set of values wherein each value or measurement has
a different weight or degree of importance.
The following is its formula:
𝑥 =
𝑥𝑊
𝑊
Where, 𝑥 is mean, W=number of measurements,
 x=measurement or value

3. Example: Below are Amaya’s subjects and the corresponding number of units
and grades she got for the previous grading period. Compute her grade point
average.
Subject Units Grade
Math1 1.5 90
Math 2 1.5 86
Computers 1.8 88
English 0.9 87
Physics 1.5 87
Solution: 𝑋 =
90∗1.5+86∗1.5+88∗1.8+87∗0.9+87∗1.5
1.5+1.5+1.8+0.9+1.5
=87.67
Amaya’s average grade is 87.67

4. Example: A student’s final scores in Mathematics, Physics, Chemistry and English are
respectively 82, 86, 90 and 70. If the respective credits received for these courses are 3,
5, 3, and 1, determine the average score.
Here the weights associated to the observations 82. 86. 90 and 70 are 3, 5, 3 and 1
x : 82 86 90 70
w : 3 5 3 1
Average
𝑥 =
𝑊𝑋
𝑊
𝑥 =
82 𝑥3+86𝑥5+90𝑥3+70𝑥1
3+5+3+1
= 84.67

5. Example: A contractor employs three types of workers, male, female and children.
To a male he pays Rs.40 per days. To a female worker Rs.32 per day and to a child
worker Rs.15 per day. The number of male, female and children workers employed
is 20, 15 and 15 respectively. Find out the average amount.
Calculation of Weighted Arithmetic Mean:
𝑥 =
𝑊𝑋
𝑊
𝑥 =
2𝑥 40+15𝑥32+15𝑥15
2+15+15
= 30.10 Rs.
Daily wages in Rs (X) No. of workers (W) WX
40 2 800
32 15 480
15 15 225
𝑊 = 50 𝑊𝑋 = 1505

6. COMBINED ARITHMETIC MEAN
If we have arithmetic mean, and the number of items of two or more than two related groups , we can
calculate the combined average of these groups by applying the formula.
𝑋1, 2,3,.... n =
𝑁1𝑥1+𝑁2𝑥2+⋯+𝑁𝑛𝑥𝑛
𝑁1+𝑁2+⋯𝑁𝑛
 𝑋1 = Arithmetic mean of the first group
 𝑋2 = Arithmetic mean of the second group
 𝑁1= Number of items in the first group
 𝑁2= Number of items in the second group

7. Example The arithmetic mean age of the first group of 80 boys is 10 years, and that
of the second group of 20 boys is 15 years. Find the arithmetic mean of the two
groups taken together.
X1, 2=
N1
X1
+ N2
X2
𝑁1
+𝑁2
N1 =80, X1=10 N2=20 X2=15
X1, 2=
80x10+ 20x15
80+20
= 11 years.

8. Example: The mean marks got by 300 students in the subject of statistics was 45.
The mean of the top 100 of them was found to be 70 and the mean of the last 100
was known to be 20. What is the mean marks of the remaining students?
Solution: 𝑋1, 2,3,.... n =
𝑁1𝑥1+𝑁2𝑥2+⋯+𝑁𝑛𝑥𝑛
𝑁1+𝑁2+⋯𝑁𝑛
45 =
100 𝑥 70 + 100 𝑥𝑋 +(100𝑥20)
300
300 x45 =100 (70+20+x)
135=90+x
X=45
Mean marks of remaining 100 students is 45.

9. Merits of Arithmetic Mean:
i. It is rigidly defined.
ii. It is easy to understand and easy to calculate.
iii. It is based upon all the observations.
iv. It is capable of further mathematical treatment.
Demerits of Arithmetic mean:
i. It cannot be determined by inspection nor it can be located graphically.
ii. Arithmetic mean cannot be used if we are dealing with qualitative characteristics which cannot be
measured quantitatively; such as, intelligence, honesty, beauty, etc.
iii. Arithmetic mean cannot be obtained if a single observation is missing or lost or is illegible unless
we drop it out and compute the arithmetic mean of the remaining values.
iv. Arithmetic mean is affected very much by extreme values.