Question

The following is the data of pocket expenditure per week of 50 students in a class. It is known that the median of the distribution is ₹120. Find the missing frequencies.

Expenditure per week (in ₹)	0 – 50	50 – 100	100 – 150	150 –200	200 –250
No. of students	7	?	15	?	3

Answer 1

Let a and b be the missing frequencies of the class 50 – 100 and class 150 – 200 respectively.

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

Expenditure per week (in ₹)	No. of students (f)	Less than Cumulative frequency (c.f.)
0 – 50	7	7
50 – 100	a	7 + a
100 – 150	15	22 + a ← Q2
150 – 200	b	22 + a + b
200 – 250	3	25 + a + b
Total	25 + a + b

Here, N = 25 + a + b
Since, N = 50
∴ 25 + a + b = 50
∴ a + b = 25 ............(i)
Given, Median = Q₂ = 120
∴ Q₂ lies in the class 100 – 150.
∴ L = 100, h = 50, f = 15, `(2"N")/4=(2xx50)/4` = 25,
c.f. = 7 + a

Q₂ = `"L"+"h"/"f"((2"N")/4-"c.f.")`

∴ 120 = `100+(50)/(15)[25-(7+"a")]`

∴ 120 – 100 = `10/3(25-7-"a")`

∴  20 = `10/3(18-"a")`
∴ `60/10` = 18 − a
∴ 6 = 18 – a
∴ a = 18 − 6 = 12
Substituting the value of a in equation (i), we get
12 + b = 25
∴ b = 25 − 12 = 13
∴ 12 and 13 are the missing frequencies of the class 50 – 100 and class 150 – 200 respectively.