Question

The following is a frequency distribution with unknown frequencies:
class. 60-70. 70-80. 80-90. 90-100 Total
frequency a/2. 3a/2. 2a. a. 50

find the value of a,hence find the frequencies.

Answer 1

For equal class intervals, we will solve by finding mid points of these classes using direct method.

CLASS

MID - POINT(xi)

TOTAL FREQUENCY(fi)

fixi

0 - 20

10

17

170

20 - 40

30

f1

30f1

40 - 60

50

32

1600

60 - 80

70

f2

70f2

80 - 100

90

19

1710

TOTAL


68 + f1 + f2

3480 + 30f1+ 70f2

We have got
Mean = 50 and N = 120 (as given in the question)
Σfi = 68 + f1 + f2 and Σfixi = 3480 + 30f1 + 70f2
∵ mean is given by

⇒  (∵ given: mean is 50)
⇒ 3400 + 50f1 + 50f2 = 3480 + 30f1 + 70f2
⇒ 50f1 – 30f1 + 50f2 – 70f2 = 3480 – 3400
⇒ 20f1 – 20f2 = 80
⇒ f1 – f2 = 4 …(i)
As given in the question, frequency(Σfi) = 120
And as calculated by us, frequency (Σfi) = 68 + f1 + f2
Equalizing them, we get
68 + f1 + f2 = 120
⇒ f1 + f2 = 120 – 68 = 52
⇒ f1 + f2 = 52 …(ii)
We will now solve equations (i) and (ii), adding them we get
(f1 + f2) + (f1 – f2) = 52 + 4
⇒ 2f1 = 56
⇒ f1 = 56/2
⇒ f1 = 28
Substitute f1 = 28 in equation (ii),
28 + f2 = 52
⇒ f2 = 52 – 28
⇒ f2 = 24
Thus, f1 = 28 and f2 = 24.