Question

Calculate upper and lower quartile from the following data:

Variable	0−10	10−20	20−30	30−40	40−50	50−60	60−70
Frequency	10	20	35	40	25	25	15

Answer 1

Class Interval	Frequency (f)	Cumulative Frequency (c.f.)
0 − 10 10 − 20	10 20	10 30
20 − 30	35	65	Q1
30 − 40	40	105
40 − 50	25	130	Q3
50 − 60 60 − 70	25 15	155 170
	N=170

$Q_1=\mathrm{Size}\mathrm{of}{\left(\frac{\mathrm{N}}{4}\right)}^{\mathrm{th}}\mathrm{item}={\left(\frac{170}{4}\right)}^{\mathrm{th}}=42.5^{\mathrm{th}}\mathrm{item}$
This lies in the class interval (20-30).

Now,
$$\begin{gathered} Q_1=l_1=\frac{{\displaystyle \frac{N}{4}}-c.f.}{f}\times i \\ \mathrm{or},Q_1=20+\frac{42.5-30}{35}\times 10 \\ \Rightarrow Q_1=23.57 \\ \mathrm{Thus},\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{lower}\mathrm{quartile}\mathrm{is}23.57. \\ {Q}_3=\mathrm{Size}\mathrm{of}3{\left(\frac{170}{4}\right)}^{\mathrm{th}}=127.5^{\mathrm{th}}\mathrm{item} \\ \mathrm{This}\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{class}\mathrm{interval}(40-50). \\ {Q}_3=l_1=\frac{3({\displaystyle \frac{N}{4})}-c.f.}{f}\times i \\ \mathrm{or},Q_3=40+\frac{127.5-105}{25}\times 10 \\ \Rightarrow Q_3=49 \\ \mathrm{Thus},\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{upper}\mathrm{quartile}\mathrm{is}49. \end{gathered}$$