Question

Calculate the quartiles, (i,e., Q1 and Q3,), deciles D3 and D7, Percentiles P10, P23 and P92 for the

following data. Also find the Quartile Range and Quartile Deviation.

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

No. of Students 8 7 5 12 28 22 8 10

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c} Class & f_i & cf \\ \hline 0-10 & 8 & 8 \\ \hdashline 10-20 & 7 & 15 \\ \hdashline 20-30 & 5 & 20 \\ \hdashline 30-40 & 12 & 32 \\ \hdashline 40-50 & 28 & 60 \\ \hdashline 50-60 & 22 & 82 \\ \hdashline 60-70 & 8 & 90 \\ \hdashline 70-80 & 10 & 100 \\ \hdashline & n=100 & \\ \end{array}$$

1.

$Q_1$  class :

Class with $(\dfrac{n }{4})^{th}$ value of the observation  in cf column

$=(\dfrac{100 }{4})^{th}$ value of the observation  in cf column

$=(25)^{th}$ value of the observation  in cf column

and it lies in the class 30-40.

$Q_1$ class : 30-40

The lower boundary point of 30-40 is 30.

$L=30$

$Q_1=L+\dfrac{\dfrac{n }{4}-cf }{f}\cdot c$

$=30+\dfrac{\dfrac{100 }{4}-20 }{12}\cdot 10$

$=34\dfrac{1 }{6}\approx34.1667$

$Q_3$  class :

Class with $(\dfrac{3n }{4})^{th}$ value of the observation  in cf column

$=(\dfrac{3\cdot100 }{4})^{th}$ value of the observation  in cf column

$=(75)^{th}$ value of the observation  in cf column

and it lies in the class 50-60.

$Q_3$ class : 50-60

The lower boundary point of 50-60 is 50.

$L=50$

$Q_3=L+\dfrac{\dfrac{3n }{4}-cf }{f}\cdot c$

$=50+\dfrac{\dfrac{300 }{4}-60 }{22}\cdot 10$

$=56\dfrac{9 }{11}\approx56.8182$

2.

$D_3$ class :

Class with $(\dfrac{3n }{10})^{th}$ value of the observation in cf column

$=(\dfrac{300 }{10})^{th}$ value of the observation in cf column

$=(30)^{th}$ value of the observation in cf column

and it lies in the class 30-40.

$D_3$ class : 30-40

The lower boundary point of 30-40 is 30.

$L=30$

$D_3=L+\dfrac{\dfrac{3n }{10}-cf }{f}\cdot c$

$=30+\dfrac{\dfrac{300 }{10}-20 }{12}\cdot 10$

$=38\dfrac{1 }{3}\approx38.3333$

$D_7$ class :

Class with $(\dfrac{7n }{10})^{th}$ value of the observation in cf column

$=(\dfrac{700 }{10})^{th}$ value of the observation in cf column

$=(70)^{th}$ value of the observation in cf column

and it lies in the class 50-60.

$D_7$ class : 50-60

The lower boundary point of 50-60 is 50.

$L=50$

$D_7=L+\dfrac{\dfrac{7n }{10}-cf }{f}\cdot c$

$=50+\dfrac{\dfrac{700 }{10}-60 }{22}\cdot 10$

$=54\dfrac{6 }{11}\approx54.5455$

3.

$P_{10}$ class :

Class with $(\dfrac{10n }{100})^{th}$ value of the observation in cf column

$=(\dfrac{1000 }{100})^{th}$ value of the observation in cf column

$=(10)^{th}$ value of the observation in cf column

and it lies in the class 10-20.

$P_{10}$ class : 10-20

The lower boundary point of 10-20 is 10.

$L=10$

$P_{10}=L+\dfrac{\dfrac{10n }{100}-cf }{f}\cdot c$

$=10+\dfrac{\dfrac{1000 }{100}-8 }{7}\cdot 10$

$=12\dfrac{6}{7}\approx12.8571$

$P_{23}$ class :

Class with $(\dfrac{23n }{100})^{th}$ value of the observation in cf column

$=(\dfrac{2300 }{100})^{th}$ value of the observation in cf column

$=(23)^{th}$ value of the observation in cf column

and it lies in the class 30-40.

$P_{23}$ class : 30-40

The lower boundary point of 30-40 is 30.

$L=30$

$P_{23}=L+\dfrac{\dfrac{23n }{100}-cf }{f}\cdot c$

$=30+\dfrac{\dfrac{2300 }{100}-20 }{12}\cdot 10$

$=32.5$

$P_{92}$ class :

Class with $(\dfrac{92n }{100})^{th}$ value of the observation in cf column

$=(\dfrac{9200 }{100})^{th}$ value of the observation in cf column

$=(92)^{th}$ value of the observation in cf column

and it lies in the class 70-80.

$P_{92}$ class : 70-80

The lower boundary point of 70-80 is 70.

$L=70$

$P_{92}=L+\dfrac{\dfrac{92n }{100}-cf }{f}\cdot c$

$=70+\dfrac{\dfrac{9200 }{100}-90 }{10}\cdot 10$

$=72$

3.

Quartile Range $IQR=Q_3-Q_1=56.8182-34.1667=22.6515$

Quartile Deviation$=\dfrac{Q_3-Q_1}{2}=\dfrac{56.8182-34.1667}{2}=11.3258$

Coefficient of Quartile deviation $=\dfrac{Q_3-Q_1}{Q_3+Q_1}$

$$=\dfrac{56.8182-34.1667}{56.8182-34.1667}=0.249$$