Question

The median of the following incomplete table is 92. Find the missing frequencies:

C.I.	30 – 50	50 – 70	70 – 90	90 – 110	110 – 130	130 – 150	Total
f	6	?	18	20	?	10	80

Answer 1

Let a and b be the missing frequencies of the class 50 – 70 and class 110 – 130 respectively.
We construct the less than cumulative frequency table as given below:

C.I.	f	Less than cumulative frequency (c.f.)
30 – 50	6	6
50 – 70	a	6 + a
70 – 90	18	24 + a
90 – 110	20	44 + a ← Q2
110 – 130	B	44 + a + b
130 – 150	10	54 + a + b
Total	N = 80

Here, N = 54 + a + b
Since, N = 80
∴ 54 + a + b = 80
∴ a + b = 26  .........(i)
Given, Median = Q₂ = 92
∴ Q₂ lies in the class 90 – 110.

∴ L = 90, h = 20, f = 20, `(2"N")/4`=`(2xx80)/4` = 40, c.f. = 24 + a

∴ Q₂ = `"L"+"h"/"f"((2"N")/4-"c.f.")`

∴ 92 = `90 + 20/20[40 - (24 + "a")]`

∴ 92 – 90 = 40 – 24 – a
∴ 2 = 16 – a
∴ a = 14
Substituting the value of a in equation (i), we get
14 + b = 26
∴ b = 26 – 14 = 12
∴ 14 and 12 are the missing frequencies of the class 50 – 70 and class 110 – 130 respectively.