Question

Find the weighted arithmetic mean for the following data.

Item	Number of items (w)	Cost of item (x)
Powder	$2$	$Rs.45$
Soap	$4$	$Rs.12$
Pen	$5$	$Rs.15$
Instruments Box	$4$	$Rs.25.50$

Answer 1

Hint: First we will first add the weights to the cost of the items by multiplying the cost by the number of items present and then we use the formula for mean. On doing some simplification we get the required answer.

Formula used: ${\text{Weighted Mean}} = \dfrac{{\sum {xw} }}{{\sum w }}$

Complete step-by-step solution:
From the given data we know the total numbers of items present in the distribution are:
$2 + 4 + 5 + 4$ Which is the sum of all the products which is: $15$.
Therefore, $\sum w = 15$
Now we have to multiply the number of items and cost of item and we get
Now, $\sum {xw} = (45 \times 2) + (12 \times 4) + (15 \times 5) + (25.50 \times 4)$
On simplifying we get:
$\Rightarrow$$\sum {xw} = 90 + 48 + 75 + 102$
Let us add we get:
$\Rightarrow$$\sum {xw} = 315$
Now to calculate the weighted mean by using the formula is:
$\Rightarrow$${\text{Weighted Mean}} = \dfrac{{\sum {xw} }}{{\sum w }}$
On substituting the values, we get:
$\Rightarrow$${\text{Weighted Mean}} = \dfrac{{315}}{{15}}$
On simplifying we get:
$\Rightarrow$${\text{Weighted Mean}} = 21$,

Hence the weighted arithmetic mean is $21$

Note: Weighted mean is a form of measuring the mean of a data with giving the values a specific weight. If the weights to all the individual values are similar, then the weighted mean found will be equal to the arithmetic mean of the data.
Elements which have more weight have a more adamant contribution than elements which have a lesser weight.
The weights of a value can be zero but they cannot be negative, they can only be positive.
The application of weighted mean is data analysis and calculus.