Question

The daily expenditure of 100 families are given below. Calculate f₁ and f₂ if the mean daily expenditure is ₹ 188.

Expenditure (in ₹)	140 – 160	160 – 180	180 – 200	200 – 220	220 – 240
Number of families	5	25	f1	f2	5

Answer 1

The given data is shown as follows:

Expenditure (in ₹)	Number of families (fi)	Class mark (xi)	fixi
140-160	5	150	750
160-180	25	170	4250
180-200	`f_1`	190	190`f_1`
200-220	`f_2`	210	210`f_2`
220-240	5	230	1150
Total	Σfi = 35 + f1 + f2		Σfixi = 6150 + 190f1 + 210f2

Sum of the frequencies = 100
⇒ `∑ f_i` = 100
⇒ 35 + `f_1 + f_2` = 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,

`barx = (sumf_i x_i)/(sumf_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.