Question

The following distribution gives the number of accidents met by $160$ workers in a factory during a month.

Number of accidents $({x_i})$	Number of workers$({f_i})$
$0$	$70$
$1$	$52$
$2$	$34$
$3$	$3$
$4$	$1$

Find the average number of accidents per worker.

Answer 1

In this question, we are given the number of accidents met by 160 workers in a month and we have been asked to find the average number of accidents per worker. For this, we will use direct mean method- mean = $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$. We are given ${f_i}$ and ${x_i}$. Multiply both of them and you will get ${f_i}{x_i}$. Add all of them and divide them with the frequency and you will get the average number of accidents per worker.

Complete step-by-step solution:
We are given the number of accidents met by 160 workers in a month. We have been asked to find the average number of accidents per worker. To find the average accidents, we will find the mean using the direct method.
First, we have to find the ${f_i}{x_i}$. Then we sum up all the ${f_i}{x_i}$.

Number of accidents $({x_i})$	Number of workers$({f_i})$	${f_i}{x_i}$
$0$	$70$	$0$
$1$	$52$	$52$
$2$	$34$	$68$
$3$	$3$	$9$
$4$	$1$	$4$
$\sum {{f_i} = 160}$	$\sum {{f_i}{x_i} = 133}$

Now, we will put the values in the formula-
Mean = $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
$\Rightarrow \bar X = \dfrac{{133}}{{160}} = 0.83$

$\therefore$ The average number of accidents per worker is $0.83.$

Note: You can also use the assumed mean method. In this method, we assume a mean and find the deviations from that assumed mean. Multiply the deviations with the frequency. Then we add ${f_i}{d_i}$ and put them in the formula. This will give us the required average. Let us solve using these steps.

Number of accidents $({x_i})$	Number of workers$({f_i})$	${d_i} = {x_i} - A$ $A = 2$	${f_i}{d_i}$
$0$	$70$	$- 2$	$- 140$
$1$	$52$	$- 1$	$- 52$
$2$= A	$34$	$0$	$0$
$3$	$3$	$1$	$3$
$4$	$1$	$2$	$2$
$\sum {{f_i} = 160}$		$\sum {{f_i}{d_i} = - 187}$

We know that $\bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$
Putting all the values,
$\Rightarrow \bar X = 2 + \left( {\dfrac{{ - 187}}{{160}}} \right)$
Solving,
$\Rightarrow \bar X = 2 - 1.17 = 0.83$
Therefore, the average number of accidents per worker is $0.83.$