Question

Calculate $Q_1$ and $Q_3$ from the following data

$\begin{array}{cc}\text{Marks}& \text{Number of}\\ & \text{Students}\\ 10& 4\\ 20& 10\\ 30& 20\\ 40& 8\\ 50& 6\\ 60& 3\end{array}$

Answer 1

Calculation of $Q_1$ and $Q_3$

$\begin{array}{ccc}\text{Marks}& \text{Number of}& \text{Cumulative}\\ & \text{Students (f)}& \text{Frequency (cf)}\\ 10& 4& 4\\ 20& 10& 14\\ 30& 20& 34\\ 40& 8& 42\\ 50& 6& 48\\ 60& 3& 51\\ & N=\sum f=51& \end{array}$

Now, $Q_1=\text{Size of}\left(\frac{N+1}{4}\right)th\text{item}$

$=\text{Size of}\left(\frac{51+1}{4}\right)\text{th item}$

= Size of 13th item and it lies in cf 14,

hence $Q_1=20\text{marks}$

and $Q_3=\text{Size of 3 times}\left(\frac{N+1}{4}\right)\text{th item}$

$=\text{Size of 3 times}\frac{(51+1)}{4}\text{th item}$

= Size of 39th item and it lies in cf 42,

hence $Q_3=40\text{marks}$