Question

The weekly profit (in rupees) of 100 shops are distributed as follows:

Profit per shop	No. of shops
0 – 1000	10
1000 – 2000	16
2000 – 3000	26
3000 – 4000	20
4000 – 5000	20
5000 – 6000	5
6000 – 7000	3

Find the limits of the profit of middle 60% of the shops.

Answer 1

To find the limits of the profit of middle 60% of the shops, we have to find P₂₀ and P₈₀.

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

Profit per shop (in rupees)	No. of shops (f)	Less than Cumulative frequency  (c.f.)
0 – 1000	10	10
1000 –  2000	16	26 ← P20
2000 –  3000	26	52
3000 – 4000	20	72
4000 –  5000	20	92 ← P80
5000 –  6000	5	97
6000 –  7000	3	100
Total	100

Here, N = 100

P₂₀ class = class containing `((20"N")/100)^"th"` observation

 ∴ `(20"N")/(100) = (20xx100)/(100)` = 20

Cumulative frequency which is just greater than (or equal to) 20 is 26.

∴ P₂₀ lies in the class 1000 – 2000.
∴ L = 1000, f = 16, c.f. = 10, h = 1000

P₂₀ = `"L"+"h"/"f"((20"N")/(100) - "c.f.")`

= `1000 + (1000)/(16)(20 - 10)`

 = `1000+125/2(10)`

= 1000 + 625
∴ P₂₀ = 1625

P₈₀ class = class containing `((80"N")/100)^"th"` observation

∴ `(80"N")/(100) = (80xx 100)/(100)` = 80

Cumulative frequency which is just greater than (or equal to) 80 is 92.

∴ P₈₀ lies in the class 4000 – 5000
∴ L = 4000, f = 20, c.f. = 72, h = 1000

P₈₀ = `"L"+"h"/"f"((80"N")/(100) - "c.f.")`

= `4000 + (1000)/(20)(80 - 72)`

= 4000 + 50(8)

= 4000 + 400
∴ P₈₀ = 4400
∴ the profit of middle 60% of the shops lies between the limits 1,625 to 4,400.