Weighted Arithmetic Mean
The weighted arithmetic mean is calculated by dividing the sum of the values (x), each multiplied by their respective weights (w), by the total sum of the weights. $$ \mu = \frac{ \sum_{i=1}^n x_i \cdot w_i}{ \sum_{i=1}^n w_i } $$
Weights can be assigned to give more importance to certain values.
Alternatively, weights can represent the absolute frequencies of the values.
In this case, we refer to the weighted arithmetic mean of a frequency distribution.
Note: The arithmetic mean is a special case of the weighted arithmetic mean, where all the weights (w) are equal to 1.
An Example
The table below shows the grades from an exam session.
The "grade" column lists the possible values xn for grades, ranging from 18 to 30.
The "students" column represents the weight wn for each grade, which is the number of students who received that particular grade.
In this case, the weights are the absolute frequencies of the grades.
To find the average, we apply the weighted mean formula.
$$ \mu = \frac{ \sum_{i=1}^n x_i \cdot w_i}{ \sum_{i=1}^n w_i } $$
$$ \mu = \frac{18 \cdot 2 + 20 \cdot 7 + 21 \cdot 4 + 22 \cdot 3 + 24 \cdot 6 + 25 \cdot 8 + 26 \cdot 4 + 27 \cdot 3 + 28 \cdot 2 + 30 \cdot 1}{2 + 7 + 4 + 3 + 6 + 8 + 4 + 3 + 2 + 1} $$
$$ \mu = \frac{36 + 140 + 84 + 66 + 144 + 200 + 104 + 81 + 56 + 30}{40} $$
$$ \mu = \frac{941}{40} $$
$$ \mu = 23.525 $$
So, the weighted arithmetic mean is 23.525.
Example 2
A university exam consists of both a written and an oral section.
The written section counts for 70% of the final grade, while the oral section makes up the remaining 30%.
A student scores 20 on the written exam and 26 on the oral exam.
The arithmetic mean of these two scores is 23.
$$ \mu = \frac{20+26}{2} = \frac{46}{2} = 23 $$
However, since the written and oral sections are weighted differently, we need to use the weighted arithmetic mean to calculate the final grade.
$$ \mu = \frac{20 \cdot 70 + 26 \cdot 30}{70 + 30} = \frac{1400 + 780}{100} = \frac{2180}{100} = 21.8 $$
The weighted arithmetic mean is 21.8.
This result is lower than the simple arithmetic mean because the written exam, which had a lower score, carries more weight.
Final Thoughts
Some key points about the weighted mean:
- The arithmetic mean is a specific case of the weighted mean, where all weights are equal to one.
And so on.
