Question

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

Weight(in kg)	Number of students
$40 - 45$	$2$
$45 - 50$	$3$
$50 - 55$	$8$
$55 - 60$	$6$
$60 - 65$	$6$
$65 - 70$	$3$
$70 - 75$	$2$

Answer 1

Hint: The formula used to find the median of a given data is as follows: $Median = l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right) \times h$
Where $l$ is the lower limit of median class, $n$ is the sum of all frequencies, $cf$ is the cumulative frequency before the median class, $f$ is the frequency of median class and $h$ is the size of median class.

Complete step-by-step answer:
The class intervals with respective cumulative frequencies can be represented as follows:

Weight(in kg)	Frequency$\left( f \right)$	Cumulative frequency$\left( {cf} \right)$
$40 - 45$	$2$	$2$
$45 - 50$	$3$	$5$
$50 - 55$	$8$	$13$
$55 - 60$	$6$	$19$
$60 - 65$	$6$	$25$
$65 - 70$	$3$	$28$
$70 - 75$	$2$	$30$
	$n = \sum {f = 30} $

From the table, we obtain $n = 30 \Rightarrow \dfrac{n}{2} = 15$
Cumulative frequency $\left( {cf} \right)$ just greater than $\dfrac{n}{2}$$\left( {i.e.,15} \right)$ is $19$, which lies in the interval $55 - 60$.
Therefore, median class=$55 - 60$
Lower limit of the median class, $l = 55$
Frequency of the median class, $f = 6$
Cumulative frequency of the class preceding the median class, $cf = 13$
Class size, $h = 5$
Therefore, $Median = l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right) \times h$
$Median = 55 + \left( {\dfrac{{15 - 13}}{6}} \right) \times 5$
$ \Rightarrow Median = 55 + \dfrac{{2 \times 5}}{6}$
$ \Rightarrow Median = 55 + \dfrac{{10}}{6}$
$ \Rightarrow Median = 56.67$
Therefore, the median weight of students is $56.67$.

Note: The cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. The last value will always be equal to the sum of all frequencies, since all frequencies will already have been added to the previous total.