Question

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

Answer 1

Hint: Mean is the average of a given set of data. Mean is also called as the arithmetic mean which is a statistical value in statistics and probability theory. The mean can be found out from the given set of data by the ratio of the summation of the given set of values to the total number of given sets of values. The arithmetic mean or the average means the one and the same which is given by the ratio of sum of given values to the total number of given values.

Complete step-by-step solution:
The arithmetic mean is given by of ‘n’ observations is given by:
$ \Rightarrow {x_n} = \dfrac{{\sum\limits_i {{x_i}} }}{n}$
$ \Rightarrow \dfrac{{\sum\limits_i {{x_i}} }}{n} = \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n}}}{n}$
Here in the given problem, the total number of workers is 60.
$\therefore n = 60$
In that 16 workers earn 3000:
\[ \Rightarrow {x_1} + {x_2} + {x_3} + \cdot \cdot \cdot + {x_{16}} = 3000 + 3000 + 3000 + \cdot \cdot \cdot + 3000\] for 16 times.
$\therefore {x_1} + {x_2} + {x_3} + \cdot \cdot \cdot + {x_{16}} = 16(3000)$
Similarly 12 workers earn 4000:
$\therefore {x_{17}} + {x_{18}} + {x_{19}} + \cdot \cdot \cdot + {x_{28}} = 12(4000)$
$ \Rightarrow $ 10 workers earn 5000:
$\therefore {x_{29}} + {x_{30}} + {x_{31}} + \cdot \cdot \cdot + {x_{38}} = 10(5000)$
$ \Rightarrow $ 8 workers earn 6000:
$\therefore {x_{39}} + {x_{40}} + {x_{41}} + \cdot \cdot \cdot + {x_{46}} = 8(6000)$
$ \Rightarrow $ 6 workers earn 7000:
\[\therefore {x_{47}} + {x_{48}} + {x_{49}} + \cdot \cdot \cdot + {x_{52}} = 6(7000)\]
$ \Rightarrow $ 4 workers earn 8000:
$\therefore {x_{53}} + {x_{54}} + {x_{55}} + {x_{56}} = 4(8000)$
$ \Rightarrow $ 3 workers earn 9000:
$\therefore {x_{57}} + {x_{58}} + {x_{59}} = 3(9000)$
$ \Rightarrow $ 1 worker earn 10000:
$\therefore {x_{60}} = 1(10000)$
$\therefore $The mean salary is given by ${x_n}$:
$ \Rightarrow {x_n} = \dfrac{{\sum\limits_{i = 1}^{60} {{x_i}} }}{{60}}$
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{60} {{x_i}} }}{{60}} = \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_{60}}}}{{60}}$
$ \Rightarrow {x_n} = \dfrac{{16(3000) + 12(4000) + 10(5000) + 8(6000) + 6(7000) + 4(8000) + 3(9000) + 1(10000)}}{{60}}$
$ \Rightarrow {x_n} = \dfrac{{48000 + 48000 + 50000 + 48000 + 42000 + 32000 + 27000 + 10000}}{{60}}$
$ \Rightarrow {x_n} = \dfrac{{305000}}{{60}}$
$ \Rightarrow {x_n} = 5083.33$
$\therefore $The mean salary is 5083.33

The mean salary of 60 workers of the factory is 5083.33

Note: While solving the arithmetic mean here, as there were multiple number of workers having the same value of observation (in this case salary is the value), the multiple number of worker’s salary is multiplied with the number of persons, as each of them has the same amount of salary and summed with the other observations to extract the value of the mean.