1. Calculate Quartile deviation from the following grouped data

X	Frequency
0	1
1	5
2	10
3	6
4	3

Solution:
Quartile deviation :

`x`	Frequency `f`	`cf`
0	1	1
1	5	6
2	10	16
3	6	22
4	3	25
---	---	---
	n = 25	--

Here, `n = 25`

`Q_1 = ((n+1)/4)^(th)` value of the observation

`=(26/4)^(th)` value of the observation

`=(6.5)^(th)` value of the observation

`=2`

---

`Q_3 = ((3(n+1))/4)^(th)` value of the observation

`=((3*26)/4)^(th)` value of the observation

`=(19.5)^(th)` value of the observation

`=3`

---

Quartile deviation `=(Q_3 - Q_1)/2=(3-2)/2=1/2=0.5`

Coefficient of Quartile deviation `=(Q_3 - Q_1)/(Q_3 + Q_1)=(3-2)/(3+2)=1/5=0.2`

---

2. Calculate Quartile deviation from the following grouped data

X	Frequency
10	3
11	12
12	18
13	12
14	3

Solution:
Quartile deviation :

`x`	Frequency `f`	`cf`
10	3	3
11	12	15
12	18	33
13	12	45
14	3	48
---	---	---
	n = 48	--

Here, `n = 48`

`Q_1 = ((n+1)/4)^(th)` value of the observation

`=(49/4)^(th)` value of the observation

`=(12.25)^(th)` value of the observation

`=11`

---

`Q_3 = ((3(n+1))/4)^(th)` value of the observation

`=((3*49)/4)^(th)` value of the observation

`=(36.75)^(th)` value of the observation

`=13`

---

Quartile deviation `=(Q_3 - Q_1)/2=(13-11)/2=2/2=1`

Coefficient of Quartile deviation `=(Q_3 - Q_1)/(Q_3 + Q_1)=(13-11)/(13+11)=2/24=0.0833`

---

3. Calculate Quartile deviation from the following grouped data

Class	Frequency
2 - 4	3
4 - 6	4
6 - 8	2
8 - 10	1

Solution:
Quartile deviation :

Class	Frequency `f`	`cf`
2 - 4	3	3
4 - 6	4	7
6 - 8	2	9
8 - 10	1	10
---	---	---
	n = 10	--

Here, `n = 10`


`Q_1` class :

Class with `(n/4)^(th)` value of the observation in `cf` column

`=(10/4)^(th)` value of the observation in `cf` column

`=(2.5)^(th)` value of the observation in `cf` column

and it lies in the class `2 - 4`.

`:. Q_1` class : `2 - 4`

The lower boundary point of `2 - 4` is `2`.

`:. L = 2`

`Q_1 = L + (( n)/4 - cf)/f * c`

`=2 + (2.5 - 0)/3 * 2`

`=2 + (2.5)/3 * 2`

`=2 + 1.6667`

`=3.6667`

---

`Q_3` class :

Class with `((3n)/4)^(th)` value of the observation in `cf` column

`=((3*10)/4)^(th)` value of the observation in `cf` column

`=(7.5)^(th)` value of the observation in `cf` column

and it lies in the class `6 - 8`.

`:. Q_3` class : `6 - 8`

The lower boundary point of `6 - 8` is `6`.

`:. L = 6`

`Q_3 = L + ((3 n)/4 - cf)/f * c`

`=6 + (7.5 - 7)/2 * 2`

`=6 + (0.5)/2 * 2`

`=6 + 0.5`

`=6.5`

---

Quartile deviation `=(Q_3 - Q_1)/2=(6.5-3.6667)/2=2.8333/2=1.4166`

Coefficient of Quartile deviation `=(Q_3 - Q_1)/(Q_3 + Q_1)=(6.5-3.6667)/(6.5+3.6667)=2.8333/10.1667=0.2787`

---

4. Calculate Quartile deviation from the following grouped data

Class	Frequency
0 - 2	5
2 - 4	16
4 - 6	13
6 - 8	7
8 - 10	5
10 - 12	4

Solution:
Quartile deviation :

Class	Frequency `f`	`cf`
0 - 2	5	5
2 - 4	16	21
4 - 6	13	34
6 - 8	7	41
8 - 10	5	46
10 - 12	4	50
---	---	---
	n = 50	--

Here, `n = 50`


`Q_1` class :

Class with `(n/4)^(th)` value of the observation in `cf` column

`=(50/4)^(th)` value of the observation in `cf` column

`=(12.5)^(th)` value of the observation in `cf` column

and it lies in the class `2 - 4`.

`:. Q_1` class : `2 - 4`

The lower boundary point of `2 - 4` is `2`.

`:. L = 2`

`Q_1 = L + (( n)/4 - cf)/f * c`

`=2 + (12.5 - 5)/16 * 2`

`=2 + (7.5)/16 * 2`

`=2 + 0.9375`

`=2.9375`

---

`Q_3` class :

Class with `((3n)/4)^(th)` value of the observation in `cf` column

`=((3*50)/4)^(th)` value of the observation in `cf` column

`=(37.5)^(th)` value of the observation in `cf` column

and it lies in the class `6 - 8`.

`:. Q_3` class : `6 - 8`

The lower boundary point of `6 - 8` is `6`.

`:. L = 6`

`Q_3 = L + ((3 n)/4 - cf)/f * c`

`=6 + (37.5 - 34)/7 * 2`

`=6 + (3.5)/7 * 2`

`=6 + 1`

`=7`

---

Quartile deviation `=(Q_3 - Q_1)/2=(7-2.9375)/2=4.0625/2=2.0312`

Coefficient of Quartile deviation `=(Q_3 - Q_1)/(Q_3 + Q_1)=(7-2.9375)/(7+2.9375)=4.0625/9.9375=0.4088`

---