Question

Calculate Q.D. from the following data:

X (less than)	10	20	30	40	50	60	70
Frequency	5	8	15	20	30	33	35

Answer 1

We construct the less than cumulative frequency table as follows:

Class	f	Less than cumulative frequency (c.f.)
0 – 10	5	5
10 – 20	3	8
20 – 30	7	15 ← Q1
30 – 40	5	20
40 – 50	10	30 ← Q3
50 – 60	3	33
60 – 70	2	35
Total	N = 35

Here, N = 35

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

∴ `"N"/4 = 35/4` = 8.75
Cumulative frequency which is just greater than (or equal to) 8.75 is 15.
∴ Q₁ lies in the class 20 – 30.
∴ L = 20, c.f. = 8, f = 7, h = 10

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

= `20 + 10/7(8.75 - 8)`

= `20 + 10/7 xx 0.75`

= 20 + 1.07
∴ Q₁ = 21.07

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

∴ `(3"N")/4 = (3 xx 35)/4` = 26.25
Cumulative frequency which is just greater than (or equal to) 26.25 is 30.
∴ Q₃ lies in the class 40 – 50.
∴ L = 40, c.f. = 20, f = 10, h = 10

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

= `40 + 10/10(26.25 - 20)`

= 40 + 6.25
∴ Q₃ = 46.25

Q.D. = `("Q"_3 - "Q"_1)/2`

= `(46.25 - 21.07)/2`

= `(25.18)/2`

= 12.59