Answer to Question #286766 in Statistics and Probability for Denevar

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 $=\dfrac{Q_3-Q_1}{Q_3+Q_1}$

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