Question

The daily expenditure of 100 families are given below. Calculate $f_1$ and $f_2$ if the mean daily expenditure is Rs. 188.
$\begin{array}{cccccc}\text{Expenditure (in Rs.)}& 140-160& 160-180& 180-200& 200-220& 220-240\\ \text{Number of}& & & & & \\ \text{families}& 5& 25& f_1& f_2& 5\end{array}$

Answer 1

The given data is shown as follows:

Expenditure (in Rs.)	Number of families ($f_i$)	Class mark ($x_i$)	($f_i{x}_i$)
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	$\sum f_i=35+f_1+f_2$		$\sum f_i{x}_i$=6150+190$f_1$+210$f_2$

Sum of the frequencies = 100

⇒ $\sum f_i$ = 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

$\overline{x}$= ∑(fi×xi)/∑fi

⇒ 188 = $\frac{6150+190f_1+210f_2}{100}$

⇒ 18800 = 6150 + 190$f_1$ + 210$f_2$

⇒ 18800 - 6150 = 190$f_1$ + 210$f_2$

⇒ 12650 = 190$f_1$ + 210(65 - $f_1$) [from (1)]

⇒ 12650 = 190$f_1$ - 210$f_1$ + 13650

⇒ 20$f_1$ = 13650 - 12650

⇒ 20$f_1$ = 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.