Question

Find the unknown frequency if 24 is the median of the following frequency distribution:
\[\begin{array}{|c|c|c|c|c|c|} \hline \text{Class-interval} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Frequency} & 5 & 25 & 25 & \text{$p$} & 7 \\ \hline \end{array}\]

Hint:

The median class is located where the cumulative frequency first exceeds $N/2$. Substitute into the median formula carefully, and solve step by step for the unknown frequency.

Answer 1

Let the total frequency be $N=5+25+25+p+7=62+p$. Median = 24, which lies in class $20$--$30$. Thus median class $=20$--$30$, with $l=20,\ h=10,\ f=25,\ c_f=30$ (cumulative frequency before 20--30). Formula: \[ \text{Median} = l + \left(\frac{\frac{N}{2}-c_f}{f}\right)h \] Substitute values: \[ 24 = 20 + \left(\frac{\frac{62+p}{2}-30}{25}\right)\times 10 \] \[ 24 = 20 + \frac{(31+\tfrac{p}{2}-30)}{25}\times 10 \] \[ 24 = 20 + \frac{(1+\tfrac{p}{2})}{25}\times 10 \] \[ 24-20 = \frac{10(1+\tfrac{p}{2})}{25} \] \[ 4 = \frac{10+5p}{25} \] \[ 100 = 10 + 5p \] \[ 5p = 90 \Rightarrow p=18 \] \[ \boxed{p=18} \]