Question

Following data gives the weight of boxes.
Calculate Q.D. for the data:

Weight (kg.)	10 – 12	12 – 14	14 – 16	16 – 18	18 – 20	20 – 22
No. of boxes	3	7	16	14	18	2
c.f.	3	10	26	40	58	60

Answer 1

Weight (kg.)	No. of boxes (f)	c.f.
10 – 12	3	3
12 – 14	7	10
14 – 16	16	26 ← Q1
16 – 18	14	40
18 – 20	18	58 ← Q3
20 – 22	2	60
Total	N = 60

Here, N = 60

Q₁ class = class containing `(("N")/4)^"th"` observation

∴ `"N"/(4) = (60)/(4)` = 15

Cumulative frequency which is just greater than (or equal to) 15 is 26.

∴ Q₁ Lies in the class 14 – 16
∴ L = 14, f = 16, c.f. = 10, h = 2

∴ Q₁ = `"L"+"h"/"f"("N"/4-"c.f.")`

= `14 + (2)/(16)(15 - 10)`

= `14 + 1/8xx5`

= 14 + 0.625
∴ Q₁ = 14.625

Q₃ class = class containing `((3"N")/4)^"th"` observation

∴  `(3"N")/(4) = (3xx60)/4` = 45

Cumulative frequency which is just greater than (or equal to) 45 is 58.

∴ Q₃ Lies in the class 18 – 20
∴ L = 18, f = 18, c.f. = 40, h = 2

∴ Q₃ = `"L"+"h"/"f"((3"N")/4-"c.f.")`

= `18 + (2)/(18)(45 - 40)`

= `18 + 1/9xx5`
= 18 + 0.5556
∴ Q₃ = 18.5556

∴ Q.D. = `("Q"_3 - "Q"_1)/2`

= `(18.5556 - 14.625)/2`

= `3.9306/2`
= 1.9653