Question

Which of the following options are true about the weighted average?

• A. The weighted average assigns certain weights to each of the individual quantities.
• B. $\text{Weighted Average}=\frac{(w_1\times n_1)+(w_2\times n_2)+\dots +(w_l\times n_l)}{w_1+w_2+\dots +w_l}$,
  
  where $w_1,w_2,\dots ,w_l$ are the weighted factors and $n_1,n_2,\dots ,n_l$ are the data points.
• C. Weighted average is the same as the average of the data set.
• D. Assigning weights would mean different levels of importance to different quantities.
  

Answer 1

Answer: (A), (B), (D)

Assigning weights would mean different levels of importance to different quantities.


The average resulting from the multiplication of each value by a weighted fcator is known as the weighted average.

Thus, the formula to calculate the weighted average is given as,

$\text{Weighted Average}=\frac{(w_1\times n_1)+(w_2\times n_2)+\dots +(w_l\times n_l)}{w_1+w_2+\dots +w_l}$,

where $w_1,w_2,\dots ,w_l$ are the weighted factors and $n_1,n_2,\dots ,n_l$ are the data points.

For example, consider the following data which depicts the average score of each group of grade $7$ students with different number of students in each group.


Here, the number of students are the weighted factors and their average scores are the corresponding data points. These can be represented as follows,


Thus, the weighted average score of grade $7$ students will be calculated as,

$\begin{array}{cc}\text{Weighted Average score}& =\frac{(w_1\times n_1)+(w_2\times n_2)+(w_3\times n_3)+(w_4\times n_4)}{w_1+w_2+w_3+w_4}\\ & =\frac{(12\times 23)+(16\times 25)+(13\times 19)+(15\times 27)}{12+16+13+15}\\ & =\frac{1328}{56}\\ & =23.71\end{array}$

Hence, the weighted average score of grade $7$ students is $23.71$

On the other hand the average of this data set will be calculated as,

$\begin{array}{cc}\text{Average}& =\frac{\text{Sum of data points}}{\text{Total number of data points}}\\ & =\frac{23+25+19+27}{4}\\ & =\frac{94}{4}\\ & =23.5\end{array}$

Thus, it can be concluded that the weighted average and the average of the data set is not same.

Moreover, the weighted average assigns certain weights to each of the individual quantities and means different levels of importance to different quantities.

So, first, second and fourth options are correct.