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	f1	190	190f1
200−220	f2	210	210f2
220−240	5	230	1150
Total	∑ fi = 35 + f1 + f2		∑ fixi = 6150 + 190f1 + 210f2

Sum of the frequencies = 100

$$\begin{gathered} \Rightarrow \sum _i{f}_i=100 \\ \Rightarrow 35+f_1+f_2=100 \\ \Rightarrow f_1+f_2=100-35 \\ \Rightarrow f_1+f_2=65 \\ \Rightarrow f_2=65-f_1....\left(1\right) \end{gathered}$$

Now, The mean of given data is given by

$$\begin{gathered} \overline{x}=\frac{{\displaystyle \sum _i}{f}_i{x}_i}{{\displaystyle \sum _i}{f}_i} \\ \Rightarrow 188=\frac{6150+190f_1+210f_2}{100} \\ \Rightarrow 18800=6150+190f_1+210f_2 \\ \Rightarrow 18800-6150=190f_1+210f_2 \\ \Rightarrow 12650=190f_1+210\left(65-f_1\right)\left[\mathrm{from}\left(1\right)\right] \\ \Rightarrow 12650=190f_1-210f_1+13650 \\ \Rightarrow 20f_1=13650-12650 \\ \Rightarrow 20f_1=1000 \\ \Rightarrow f_1=50 \end{gathered}$$

If f₁ = 50, then f₂ = 65 − 50 = 15

​Thus, the value of f₁ is 50 and f₂ is 15.