Question

The distribution below gives the weights of $30$ students of a class. Find the median weight of the students.

Weight (in kg)	$40-45$	$45-50$	$50-55$	$55-60$	$60-65$	$65-70$	$70-75$
No. of students	$2$	$3$	$8$	$6$	$6$	$3$	$2$

Answer 1

Step 1: Creating cumulative frequency table

First, prepare a cumulative frequency table as below:

Weight (in kg)	No. of students (Frequency)	Cumulative Frequency ($cf$)
$40-45$	$2$	$2$
$45-50$	$3$	$2+3=5$
$50-55$	$8$	$5+8=13$
$55-60$	$6$	$13+6=19$
$60-65$	$6$	$19+6=25$
$65-70$	$3$	$25+3=28$
$70-75$	$2$	$28+2=30$

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 $n=30$

$\Rightarrow \frac{30}{2}=15$

The observation $15$ lies in the class $55-60$.

So, we have

$l=55,c=13,f=6,h=5$

$$\begin{gathered} \Rightarrow Median=55+\left(\frac{15-13}{6}\right)\times 5 \\ \Rightarrow Median=55+1.67 \\ \therefore Median=56.67 \end{gathered}$$

Hence, the median weight of the students is $56.67kg$.