Question

The daily expenditure of 100 families are given below. Calculate `f_1` and `f_2` if the mean daily expenditure is ₹ 188.

Expenditure (in Rs)	140-160	160-180	180-200	200-220	220-240
Number of families	5	25	`f_1`	`f_2`	5

Answer 1

The given data is shown as follows:

Expenditure(in Rs)	Number of families `(f_i)`	class mark `(x_i)`	`f_ix_i`
140-160	5	150	750
160-180	25	170	4250
180-200	`f_i`	190	190`f_i`
200-220	`f_2`	210	210`f_2`
220-240	5	230	1150
Total	`sum f_i = 35 + f_1 + f_2`		`sum f_i x_i = 6150+190f_1+210f_2`

Sum of the frequencies = 100
⇒ `∑ _i f_i` = 100
⇒ 35 + `f_1` + 𝑓2 = 100
⇒` f_1 + f_2` = 100 – 35
⇒ `f_1 + f_2` = 65

⇒ `f_2 = 65 – f_1`           ……..(1)
Now, the mean of the given data is given by,

x =`(sum_(i) f_i x_i)/(sum _(i) f_i )`

⇒ 188 =`(6150+190f_1+210f_2)/100`
`⇒ 18800 = 6150 + 190f_1 + 210f_2`
`⇒ 18800 – 6150 = 190f_1 + 210f_2`
`⇒ 12650 = 190f_1 + 210(65 – f_1) `             [from (1)]
`⇒ 12650 = 190f_1 – 210f_1 + 13650`
`⇒ 20f_1 = 13650 - 12650`
`⇒ 20f_1 = 1000`
`⇒ f_1 = 50`
`If f_1 = 50, then f_2 = 65 – 50 = 15`
`"Thus, the value of"  f_1   is  50 and f_2   is   15.`