Calculate Lower Quartile and Upper Quartile from the following data :

$\begin{array}{ccccccccc}\text{X}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ \text{f}& 15& 20& 25& 24& 12& 31& 71& 52\end{array}$

To calculate Lower Quartile and Upper Quartile

$\begin{array}{ccc}\text{X}& f& c.f\\ 0-10& 15& 15\\ 10-20& 20& 35\\ 20-30& 25& 60\\ 30-40& 24& 84\\ 40-50& 12& 96\\ 50-60& 31& 127\\ 60-70& 71& 198\\ 70-80& 52& 250\end{array}$

Lower Quartile $=Q_1$

$\frac{N}{4}=\frac{250}{4}=62.5th\text{item}$

$Q_1$ lies in class 30 - 40

$Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times i$

$Q_1=30+\frac{\frac{250}{4}-60}{24}\times 10=31.04$

Upper Quartile $=Q_3$

$\frac{3N}{4}=\frac{3\left(250\right)}{4}=187.5th\text{item}$

$Q_3$ lies in class 60 - 70

$Q_3=l_1+\frac{\frac{3N}{4}-c.f}{f}\times i$

$Q_3=30+\frac{\frac{3\times 250}{4}-127}{71}\times 10=68.52$