The daily expenditure of 100 families are given below- Calculate f1 and f2 if the mean daily expenditure is Rs- 188- - Expenditure in Rs- 140-160 160-180 180-200 200-220 220-240- Number of- families 5 25 f1 f2 5- -

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Direct Method of Finding Mean

The daily exp...

Question

The daily expenditure of 100 families are given below. Calculate $f_{1}$ and $f_{2}$ if the mean daily expenditure is Rs. 188.
$\begin{matrix} Expenditure (in Rs.) & 140 - 160 & 160 - 180 & 180 - 200 & 200 - 220 & 220 - 240 Number of families & 5 & 25 & f_{1} & f_{2} & 5 \end{matrix}$

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Solution

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

$¯ x$ = ∑(fi×xi)/∑fi

⇒ 188 = $\frac{6150 + 190 f_{1} + 210 f_{2}}{100}$

⇒ 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.

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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 $f_{2}$

The daily expenditure of 100 families are given below. Calculate f1 and f2 if the mean daily expenditure is Rs. 188. Expenditure (in Rs.)140−160160−180180−200200−220220−240Number offamilies525f1f25

The daily expenditure of 100 families are given below. Calculate $f_{1}$ and $f_{2}$ if the mean daily expenditure is Rs. 188.
$\begin{matrix} Expenditure (in Rs.) & 140 - 160 & 160 - 180 & 180 - 200 & 200 - 220 & 220 - 240 Number of families & 5 & 25 & f_{1} & f_{2} & 5 \end{matrix}$