Ex.14.1 Q2 Statistics Solution - NCERT Maths Class 10
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 \(\rm Rs.545.20\)