Question

The following is the distribution of 160 Workers according to the wages in a certain factory:

Wages more than (in ₹)	No. of workers
8000	160
9000	155
10000	137
11000	91
12000	57
13000	23
14000	10
15000	1
16000	0

Determine the values of all quartiles and interpret the results.

Answer 1

The given table is a more than cumulative frequency. We transform the given table into less than cumulative frequency.

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

Wages (in ₹)	No. of workers (f)	Less than Cumulative frequency (c.f.)
8000 – 9000	160 – 155 = 5	5
9000 – 10000	155 –  137 = 18	23
10000 – 11000	137 –  91 = 46	69 ← Q1
11000 – 12000	91 –  57 = 34	103 ← Q2
12000 – 13000	57 –  23 = 34	137 ← Q3
13000 – 14000	23 –  10 = 13	150
14000 – 15000	10 –  1 = 9	159
15000 – 16000	1 –  0 = 1	160
16000 – 17000	0	160
Total	160

Here, N = 160
∴ Q₁ class = class containing `("N"/4)^"th"` observation

∴ `"N"/4=160/4` = 40

Cumulative frequency which is just greater than (or equal to) 40 is 69.

∴ Q₁ lies in the class 10000 – 11000
∴ L = 10000, f = 46, c.f. = 23, h = 1000

Q₁ = `"L"+"h"/"f"("N"/4-"c.f.")`

= `10000 + (1000)/(46)(40 - 23)`

= `10000+1000/46(17)`

= `10000+17000/46`

= 10000 + 369.57
= 10369.57
Q₂ class = class containing `((2"N")/4)^"th"` observation

∴ `(2"N")/4=(2xx160)/4` = 80

Cumulative frequency which is just greater than (or equal to) 80 is 103
∴ Q₂ lies in the class 11000 – 12000
∴ L = 11000, f = 34, c.f. = 69, h = 1000

Q₂ = `"L"+"h"/"f"("2N"/4 - "c.f.")`

= `11000 + (1000)/(34)(80 - 69)`

= `11000+1000/34(11)`

= `11000+11000/34`

=  11000 + 323.529
= 11323.529

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

`(3"N")/(4) = (3(160))/(4)` = 120

Cumulative frequency which is just greater than (or equal to) 120 is 137.
∴ Q₃ lies in the class 12000 – 13000
∴ L = 12000, f = 34, c.f. = 103; h = 1000

Q₃ = `"L"+"h"/"f"((3"N")/4 - "c.f.")`

= `12000 + (1000)/(34)(120 - 103)`

= `12000+1000/34(17)`

= `12000+1000/2`

= 12000 + 500
= 12500
∴ The quartiles are
Q₁ =  Rs.10369.57
Q₂ = Rs. 11323.529
Q₃ = Rs. 12500
Q₁ < Q₂ < Q_3.