Question

Weight (kg)	40 – 45	45 – 50	50 – 55	55 – 60	60 – 65	65 –70	70 – 75	75 – 80
No. of person	4	15	20	30	20	10	8	4

For above data, find all quartiles and number of persons weighing between 57 kg and 72.

Answer 1

We construct the less than cumulative frequency table as given below:

Weight (kg)	No. of persons (f)	Less than cumulative frequency (c.f.)
40 – 45	4	4
45 – 50	15	19
50 – 55	20	39 ← Q1
55 – 60	30	69 ← Q2, Px
60 – 65	20	89 ← Q3
65 – 70	10	99
70 – 75	8	107 ← Py
75 – 80	4	111
Total	N = 111

Here, N = 111

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

∴ `"N"/4 = 111/4` = 27.75
Cumulative frequency which is just greater than (or equal) to 27.75 is 39.
∴ Q₁ lies in the class 50 – 55.
∴ L = 50, h = 5, f = 20, c.f. = 19

∴ Q₁ = `L + h/f(N/4 - c.f.)`

= `50 + 5/20(27.75- 19)`

= `50 + 5/20 xx 8.75`

= `50 + (43.75)/20`

= 50 + 2.1875
= 52.1875

Q₂ class = class containing `((2N)/4)^"th"` observation

∴ `(2N)/4 = (2 xx 111)/4` = 55.5
Cumulative frequency which is just greater than (or equal) to 55.5 is 69.
∴ Q₂ lies in the class 55 – 60.
∴ L = 55, h = 5, f = 30, c.f. = 39

∴ Q₂  = `L + h/f((2N)/4 - c.f.)`

= `55 + 5/30 (55.5 - 39)`

= `55 + 1/6 xx 16.5`

= 55 + 2.75
= 57.75

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

∴ `(3N)/4 = (3 xx 111)/4` = 83.25
Cumulative frequency which is just greater than (or equal) to 83.25 is 89.
∴ Q₃ lies in the class 60 – 65.
∴ L = 60, h = 5, f = 20, c.f. = 69

∴ Q₃  = `L + h/f((3N)/4 - c.f.)`

= `60 + 5/20 (83.25 - 69)`

= `60 + 1/4 xx 14.25`

= 60 + 3.5625
= 63.5625
In order to find the number of persons between 57 kg and 72 kg,
We need to find x in Px, where Px = 57 kg and y in Py, where Py = 72 kg
Then (y – x) would be the % of persons weighing between 57 kg and 72 kg
P_x = 57

∴ `L + h/f((x xx N)/100 - c.f.)` = 57

∴ `55 + 5/30 (1.11x - 39)` = 57

∴ `1/6(1.11x - 39)` = 57

∴ 1.11x – 39 = 12
∴ 1.11x = 51
∴ x = 45.95
∴ P_x = 72

∴ `L + h/f((y xx N)/100 - c.f.)` = 72

∴ `70 + 5/8 (1.11y - 99)` = 72

∴ 0.625(1.11y − 99) = 72

∴ 1.11y – 99 = 3.2
∴ 1.11y = 102.2
∴ y = 92.07
∴ % of people weighing between 57 kg and 72 kg = 92.07 – 45.95 = 46.12 %
∴ No. of people weighing between 57 kg and 72 kg = 111 × 46.12% = 51.1932 ≈ 51