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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 |
Solution
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.`
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