Math, asked by dileepkumarvadakuttu, 19 days ago

The daily wages of 80 workers of a factory are shown below. Find the mean of the following data by using a suitable method Amount (Rs) 300-400 400-500 500-600 600-700 700-800 No of workers 14 18 23 15 10​

Answers

Answered by mathdude500
40

\large\underline{\sf{Solution-}}

Given data is

\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \small\boxed{\begin{array}{c |c} \tt{Amount} & \tt{Number \: of \: workers} \\ \dfrac{\qquad\qquad}{ \sf 300-400} &\dfrac{\qquad\qquad}{ \sf 14} & \\ \dfrac{\qquad\qquad}{ \sf 400-500} &\dfrac{\qquad\qquad}{ \sf 18} & \\ \dfrac{\qquad\qquad}{ \sf 500-600} &\dfrac{\qquad\qquad}{ \sf 23} & \\ \dfrac{\qquad\qquad}{ \sf 600-700} &\dfrac{\qquad\qquad}{ \sf 15} & \\ \dfrac{\qquad\qquad}{ \sf 700-800} &\dfrac{\qquad\qquad}{ \sf 10} & &\end{array}} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}

Now, frequency distribution table for calculations of mean using Step Deviation Method.

\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c|c|c}\sf Class\: interval&\sf Frequency\: (f_i)&\sf \: midvalue \: (x_i)&\sf \: u_i&\sf \: f_iu_i\\\frac{\qquad  \qquad}{}&\frac{\qquad  \qquad}{}\\\sf 300 - 400&\sf 14&\sf350&\sf - 2&\sf - 28\\\\\sf 400 - 500 &\sf 18&\sf450&\sf - 1&\sf - 18\\\\\sf 500-600 &\sf 23 &\sf550 - A&\sf0&\sf0\\\\\sf 600 - 700&\sf 15&\sf650&\sf1&\sf15\\\\\sf 700-800&\sf 10&\sf750&\sf2&\sf20\\\frac{\qquad}{}&\frac{\qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}\end{gathered}

So, we get from above table

\rm \: A = 550 \\

\rm \: h = 100 \\

\rm \: \displaystyle\sum f_i \:  =  \: 80 \\

\rm \: \displaystyle\sum f_iu_i \:  =  \:  - 11 \\

Now, we know, Mean using Step Deviation Method is given by

\rm \: Mean = A + \dfrac{\displaystyle\sum f_i \: u_i}{\displaystyle\sum f_i}  \times h \\

So, on substituting the values, we get

\rm \: Mean \:  =  \: 550 \:  +  \: \dfrac{ - 11}{80} \times 100

\rm \: Mean \:  =  \: 550 \:   -   \: 13.75 \\

\rm\implies \:Mean \:  =  \: 486.25 \\

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Additional Information :-

1. Mean using Direct Method is given by

\boxed{\sf{  \: \: \rm \: Mean =  \dfrac{\displaystyle\sum f_i \: x_i}{\displaystyle\sum f_i}  \:  \: }} \\

2. Mean using Short Cut Method is given by

\boxed{\sf{  \: \: \rm \: Mean = A +  \dfrac{\displaystyle\sum f_i \: d_i}{\displaystyle\sum f_i}  \:  \: }} \\

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