2. Median - Ungrouped frequency distribution

Find the median of the following data.

 

 

Solution:

 

Let us arrange the marks in ascending order and find the cumulative frequency.

 

 

Therefore, the total frequency $N = 30$.

 

Since $N$ is even, the median can be determined using the formula $\left(\frac{(\frac{N}{2}^{th}) \text{observation} + (\frac{N}{2}+1)^{th} \text{observation}}{2} \right)$

 

Substituting the known values, we get:

 

Median $= \left(\frac{(\frac{30}{2}^{th}) \text{observation} + (\frac{30}{2}+1)^{th} \text{observation}}{2} \right)$

 

$= \left(\frac{15^{th} \text{observation} + 16^{th} \text{observation}}{2} \right)$

 

If the marks of the students are arranged in ascending order, then the $15^{th}$ and $16^{th}$ terms would be in the middle value. From the cumulative frequency distribution, let us select the marks near the $15^{th}$ and $16^{th}$ terms. Thus, the marks would be $75$.

 

Substituting these values in the median, we have:

 

Median $= \frac{75+75}{2}$ $=75$

 

Therefore, the median of the given data is $75$.