Question

Find the mean salary of 60 workers of a factory from the following table;

Salary (in Rs.)	Number of workers
3000	16
4000	12
5000	10
6000	8
7000	6
8000	4
9000	3
10000	1
Total	60

Answer 1

Hint: We start solving the problem by recalling the definition of weighted mean as the ratio of the weighted sum of observations to the sum of weights. We use this definition and use the formula of mean salary as $\dfrac{\sum\limits_{{}}^{{}}{\left( \text{no}\text{. of workers} \right)\times \left( salary \right)}}{\sum\limits_{{}}^{{}}{\left( \text{no}\text{. of workers} \right)}}$. We then substitute the salary and the no. of workers in the formula and make necessary calculations to get the required result.

Complete step by step answer:
According to the problem, we need to find the mean salary of the 60 workers of a factory from the following table;

Salary (in Rs.)	Number of workers
3000	16
4000	12
5000	10
6000	8
7000	6
8000	4
9000	3
10000	1
Total	60

We solve this problem by using the weighted mean formula as the number of workers is not same for each salary.
Let us recall the definition of weighted mean. We know that the weighted mean is defined as the ratio of the weighted sum of observations to the sum of weights.
In short weighted mean = $\dfrac{\sum\limits_{i=1}^{n}{{{w}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{n}{{{w}_{i}}}}$,
Where, ${{w}_{i}}$ = weight of the observation related to ${{x}_{i}}$,
${{x}_{i}}$ = the value of the observation.
Using this formula, we get mean salary = $\dfrac{\sum\limits_{{}}^{{}}{\left( \text{no}\text{. of workers} \right)\times \left( salary \right)}}{\sum\limits_{{}}^{{}}{\left( \text{no}\text{. of workers} \right)}}$.
So, we get mean salary = $\dfrac{\left( 3000\times 16 \right)+\left( 4000\times 12 \right)+\left( 5000\times 10 \right)+\left( 6000\times 8 \right)+\left( 7000\times 6 \right)+\left( 8000\times 4 \right)+\left( 9000\times 3 \right)+\left( 10000\times 1 \right)}{60}$.
$\Rightarrow $ Mean salary = $\dfrac{48000+48000+50000+48000+42000+32000+27000+10000}{60}$.
$\Rightarrow $ Mean salary = $\dfrac{305000}{60}$.
$\Rightarrow $ Mean salary = $5083.33$.
So, we have found the mean salary of the 60 workers as Rs. 5083.33.

So The mean salary of the 60 workers is Rs. 5083.33.

Note: We should not take the normal arithmetic mean for salaries in this case as the no. of workers is different for each salary. We should know that there are different types of means which will be used as per the requirement of the problem. We should not make calculation mistakes while solving this problem. We can also expect problems to find the mathematical average of the salaries without giving information about the number of workers.