Question

The daily wages of 50 employees in an organization are given below:Daily wages (in $) X 100 - 150 150 - 200 200 - 250 250 - 300 Fr 12 13 17 8 Find the mean(short cut method) and median daily wages.

Answer 1

Answer:

The mean of the daily wages is $ 196.

The median of the daily wages is $ 200.

Step-by-step-explanation:

We have given the distribution of the wages of some employees.

We have to find the mean and median of the daily wages.

$\displaystyle\begin{array}{|c|c|c|c|}\cline{1-4}\bf\:Class\:(\:Daily\:wages\:in\:\ \:) & \bf\:Class\:marks\:(\:x_i\:) & \bf\:Frequency\:(\:f_i\:) & \bf\:x_i\:.\:f_i\\\cline{1-4}\sf\:100\:-\:150 & \sf\:125 & \sf\:12 & \sf\:1500\\\cline{1-4}\sf\:150\:-\:200 & \sf\:175 & \sf\:13 & \sf\:2275\\\cline{1-4}\sf\:200\:-\:250 & \sf\:225 & \sf\:17 & \sf\:3825\\\cline{1-4}\sf\:250\:-\:300 & \sf\:275 & \sf\:8 & \sf\:2200\\\cline{1-4} & & \sf\:\sum\:f_i\:=\:50 & \sf\:\sum\:x_i\:.\:f_i\:=\:9800\\\cline{1-4}\end{array}$

Now, we know that,

$\displaystyle\pink{\sf\:Mean\:\overline{X}\:=\:\dfrac{\sum\:x_i\:.\:f_i}{\sum\:f_i}}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\dfrac{980\cancel{0}}{5\cancel{0}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{980}{5}}\\\\\\\implies\boxed{\red{\sf\:Mean\:\overline{X}\:=\:196\:\ }}$

The mean of the daily wages is $ 196.

$\rule{200}{1}$

Now, we have to find the median of the given distribution of the daily wages.

$\displaystyle\begin{array}{|c|c|c|}\cline{1-3}\bf\:Class\:(\:Daily\:wages\:in\:\ \:) & \bf\:Frequency\:(\:f\:) & \bf\:Cumulative\:frequency\:(\:cf\:)\\\cline{1-3}\sf\:100\:-\:150 & \sf\:12 & \sf\:12\\\cline{1-3}\sf\:150\:-\:200 & \sf\:13 & \sf\:25\:\rightarrow\:cf\\\cline{1-3}\boxed{\sf\:200\:-\:250\:\rightarrow\:Median\:class} & \sf\:17\:\rightarrow\:f & \sf\:42\\\cline{1-3}\sf\:250\:-\:300 & \sf\:8 & \sf\:50\\\cline{1-3}& \sf\:N\:=\:\sum\:f\:=\:50 & \\\cline{1-3}\end{array}$

Now,

$\displaystyle\bullet\sf\:Lower\:class\:limit\:of\:median\:class\:(\:L\:)\:=\:200\\\\\\\bullet\sf\:Sum\:of\:frequencies\:(\:N\:)\:=\:50\\\\\\\bullet\sf\:Class\:interval\:of\:median\:class\:(\:h\:)\:=\:50\\\\\\\bullet\sf\:Frequency\:of\:median\:class\:(\:f\:)\:=\:17\\\\\\\bullet\sf\:Cumulative\:frequency\:of\:class\:preceiding\:median\:class\:(\:cf\:)\:=\:25$

Now, we know that,

$\displaystyle\pink{\sf\:Median\:=\:L\:+\:\left(\:\dfrac{\dfrac{N}{2}\:-\:cf}{f}\:\right)\:\times\:h}\sf\:\:\:-\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:Median\:=\:200\:+\:\left(\:\dfrac{\cancel{\dfrac{50}{2}}\:-\:25}{17}\:\right)\:\times\:50\\\\\\\implies\sf\:Median\:=\:200\:+\:\left(\:\dfrac{25\:-\:25}{17}\:\right)\:\times\:50\\\\\\\implies\sf\:Median\:=\:200\:+\:\dfrac{0}{17}\:\times\:50\\\\\\\implies\sf\:Median\:=\:200\:+\:0\:\times\:50\\\\\\\implies\sf\:Median\:=\:200\:+\:0\\\\\\\implies\boxed{\red{\sf\:Median\:=\:200\:\ }}$

The median of the daily wages is $ 200.