# Ex.14.1 Q2 Statistics Solution - NCERT Maths Class 10

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## Question

Consider the following distribution of daily wages of $$50$$ workers of a factory.

 Daily wages (in Rs) $$500 –520$$ $$520 – 540$$ $$540 –560$$ $$560 – 580$$ $$580 – 600$$ Number of workers $$12$$ $$14$$ $$8$$ $$6$$ $$10$$

Find the mean daily wages of the workers of the factory by using an appropriate method.

## Text Solution

What is known?

Distribution of daily wages of $$50$$ workers of a factory is given-

What is unknown?

The mean daily wages of the workers of the factory.

Reasoning:

We will use Assumed Mean Method to solve this questionbecause the data given is large.

Sometimes when the numerical values of $$x_i$$ and $$f_i$$ are large, finding the product of $$x_i$$ and $$f_i$$ becomes tedious.We can do nothing with the $$f_i’s$$, but we can change each xi to a smaller number so that our calculations become easy.Now we have to subtract a fixed number from each of these $$x_i’s$$.

The first step is to choose one among the $$x_i’s$$ as the assumed mean, and denote it by $$‘a’.$$ Also, to further reduce our calculation work, we may take ‘a’ to be that $$x_i$$ which lies in the centre of \begin{align}{x} _1, {x} _ 2, \ldots, {x_n}\end{align}. So, we can choose $$a.$$

The next step is to find the difference $$d_i$$ between a and each of the $$x_i$$’s, that is, the deviation of ‘a’ from each of the $$x_i$$'s i.e., $$d_i = x_i – a$$

The third step is to find the product of $$d_i$$ with the corresponding$$f_i$$, and take the sum of all the $${f}_{{i}} {d}_{{i}}^{\prime} {s}$$

Now put the values in the below formula

\begin{align}\operatorname{Mean} \,\,(\overline{{x}})= {a}+\left(\frac{\Sigma {f}_{{i}} {u}_{{i}}}{\Sigma {f}_{{i}}}\right) \end{align}

Steps:

We know that,

class mark ($$x_i$$) \begin{align}={\frac{\text { upper limit+lower limit }}{2}} \end{align}

Taking assumed mean, a = 550

 Daily wages (in Rs) No of workers $$(f_i)$$ $$(X_i)$$ $$d_i = x_i -150$$ $$f_iu_i$$ $$500-520$$ $$12$$ $$510$$ $$- 40$$ $$- 480$$ $$520-540$$ $$14$$ $$530$$ $$- 20$$ $$- 280$$ $$540-560$$ $$8$$ $$550 (a)$$ $$0$$ $$0$$ $$560-580$$ $$6$$ $$570$$ $$20$$ $$120$$ $$580-600$$ $$10$$ $$590$$ $$40$$ $$400$$ $$\Sigma f_i=50$$ $$\Sigma f_id_i= -240$$

It can be observed from the table,

\begin{align} \Sigma f_{i} &=50 \\ \Sigma f_{i} u_{i} &=-240 \end{align}

\begin{align}{{{\rm Mean }}\,\,(\overline {{x}} ) } &={{{a}} + \left( {\frac{{{{ }}\Sigma {{f_iu_i}}}}{{\Sigma {{f_i}}}}} \right){{h}}}\\{}&{ = 550 + \left( {\frac{{ - 240}}{{50}}} \right)}\\{}&{ = 550 - \frac{{24}}{5}}\\&=550-4.8\\ {}&{ = 545.2}\end{align}

Thus, the mean daily wages of the workers of the factory is $$\rm Rs.545.20$$

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