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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