Question

Following data gives the age distribution of 240 employees of a firm. Calculate Q.D. of the distribution:

Age (in years)	20 – 25	25 – 30	30 – 35	35 – 40	40 – 45	45 – 50
No. of employees	30	40	60	50	46	14

Answer 1

We construct the less than cumulative frequency table as follows:

Age (In years)	No. of employees (f)	Less than cumulative frequency (c.f.)
20 – 25	30	30
25 – 30	40	70 ← Q1
30 – 35	60	130
35 – 40	50	180 ← Q3
40 – 50	46	226
45 – 50	14	240
Total	N = 240

Here, N = 240

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

∴ `"N"/(4) = (240)/(4)` = 60

Cumulative frequency which is just greater than (or equal to) 60 is 70.

∴ Q₁ Lies in the class 25 – 30.

∴ L = 25, f = 40, c.f. = 30, h = 5

Q₁ = `"L"+"h"/"f"("N"/4-"c.f.")`

∴ Q₁ = `25 + (5)/(40)(60 - 30)`

∴ Q₁ = `25 + (1)/(8)(30)`

∴ Q₁ = 25 + 3.75

∴ Q₁ = 28.75

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

∴ `(3"N")/4=(3xx240)/4` = 180

Cumulative frequency which is just greater than (or equal to) 180 is 180.

∴ Q₃ Lies in the class 35 – 40

∴ L = 35, f = 50, c.f. = 130, h = 5

Q₃ = `"L"+"h"/"f"((3"N")/4-"c.f.")`

∴ Q₃  = `35 + (5)/(50)(180 - 130)`

∴ Q₃  = `35 + (1)/(10)xx50`

∴ Q₃  = 35 + 5

∴ Q₃ = 40

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

∴ Q.D. = `(40 - 28.75)/(2)`

∴ Q.D. = `(11.25)/(2)`

∴ Q.D. = 5.625