For the following data, find the weighted mean.

​​$$\begin{array}{c|cccccc}X_i & 4 & 3 & 5 & 7 & 2 & 1 \\ W_i & 1 & 0 & 2 & 1 & 3 & 5 \\ \end{array}$$​​

Correct option is A

Given:

​$$X_i = [4, 3, 5, 7, 2, 1], \\ \ \\ W_i = [1, 0, 2, 1, 3, 5]$$​​

Formula Used:

​$$∑Wi\text{Weighted Mean} = \frac{\sum (X_i \cdot W_i)}{\sum W_i} ​$$​​

Solution:

Calculating $$X_i \cdot W_i$$​ ​ for each i:

​$$[4 \cdot 1, \, 3 \cdot 0, \, 5 \cdot 2, \, 7 \cdot 1, \, 2 \cdot 3, \, 1 \cdot 5] = [4, 0, 10, 7, 6, 5]$$​​

Sum of $$X_i \cdot W_i​:$$​​

​$$\sum (X_i \cdot W_i) = 4 + 0 + 10 + 7 + 6 + 5 = 32$$​​

Sum of $$W_i​:$$​​

​$$\sum W_i = 1 + 0 + 2 + 1 + 3 + 5 = 12$$​​

the weighted mean:

​$$\text{Weighted Mean} = \frac{32}{12} = 2.67 \, (\text{rounded to two decimal places})$$​​