Question

Calculate Q.D. of the following data:

Height of plants (in feet)	2 – 4	4 – 6	6 – 8	8 – 10	10 – 12	14 – 14	14 – 16
No. of plants	15	20	25	12	18	13	17

Answer 1

We construct the less than cumulative frequency table as follows:

Height of plants (in feet)	No. of plants (f)	Less than cumulative frequency (c.f.)
2 – 4	15	15
4 – 6	20	35 ← Q1
6 – 8	25	60
8 – 10	12	72
10 – 12	18	90 ← Q3
12 – 14	13	103
14 – 16	17	120
Total	N = 120

Here, N = 120

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

∴ `"N"/4 = 120/4` = 30
Cumulative frequency which is just greater than (or equal to) 30 is 35.
∴ Q₁ lies in the class 4 – 6
∴ L = 4, c.f. = 15, f = 20, h = 2

∴ Q₁ = `"L" + "h"/"f"("N"/4 - "c.f.")`

= `4 + 2/20 (30 - 15)`

= `4 + 1/10 xx 15`

= 4 + 1.5
= 5.5

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

∴ `(3"N")/4 = (3 xx 120)/4` = 90
Cumulative frequency which is just greater than (or equal to) 90 is 90.
∴ Q₃ lies in the class 10 – 12
∴ L = 10, c.f. = 72, f = 18, h = 2

∴ Q₃ = `"L" + "h"/"f"((3"N")/4 - "c.f.")`

= `10 + 2/18 (90 - 72)`

= `10 + 2/18 xx 18`

= 10 + 2
∴ Q₃ = 12

∴ Q.D. = `("Q"_3 - "Q"_1)/2`

= `(12 - 5.5)/2`

= `(6.5)/2`

= 3.25