Question

The daily wages of $160$ workers in a building project are given below:

Wages(in Rs.)	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
No. of workers	$12$	$20$	$30$	$38$	$24$	$16$	$12$	$8$

Find the median wage of the workers.

Answer 1

Step 1: Creating cumulative frequency table

First, prepare a cumulative frequency table as below:

Wages (in Rs.)	No. of workers (Frequency)	Cumulative Frequency ($cf$)
$0-10$	$12$	$12$
$10-20$	$20$	$12+20=32$
$20-30$	$30$	$32+30=62$
$30-40$	$38$	$62+38=100$
$40-50$	$24$	$100+24=124$
$50-60$	$16$	$124+16=140$
$60-70$	$12$	$140+12=152$
$70-80$	$8$	$152+8=160$

Step 2: Calculating the median

The formula for the median is,

$Median=l+\left(\frac{{\displaystyle \frac{\mathrm{n}}{2}-\mathrm{c}}}{\mathrm{f}}\right)\times h$

Here,

$l$ is the lower limit of the median class,

$n$ is the number of observations,

$c$ is the cumulative frequency of the class preceding the median class,

$f$ is the frequency of the median class, and

$h$ is the class size

Here $n=160$

$\Rightarrow \frac{160}{2}=80$

The observation $80$ lies in the class $30-40$.

So, we have

$l=30,c=62,f=38,h=10$

$$\begin{gathered} \Rightarrow Median=30+\left(\frac{80-62}{38}\right)\times 10 \\ \Rightarrow Median=30+4.74 \\ \therefore Median=34.74 \end{gathered}$$

Hence, the median wage of the workers is $Rs34.74$.