Question

Find the missing frequencies $f_1$ and $f_2$ if the mean of the following data is $78$ and total frequency is $50$.

Class Interval	50-60	60-70	70-80	80-90	90-100	Total
Frequency	8	$f_1$	12	$f_2$	13	50

• A. $f_1=4$ and $f_2=13$
• B. $f_1=5$ and $f_2=12$
• C. $f_1=6$ and $f_2=11$
• D. $f_1=7$ and $f_2=10$

Answer 1

The correct option is C $f_1=6$ and $f_2=11$

$CI$	$f_i$	$x_i$	$f_i{x}_i$
$50-60$	$8$	$55$	$440$
$60-70$	$f_1$	$65$	$65f_1$
$70-80$	$12$	$75$	$900$
$80-90$	$f_2$	$85$	$85f_2$
$90-100$	$13$	$95$	$1235$
	$\sum f_i=33+f_1+f_2$		$\sum f_i{x}_i=2575+65f_1+85f_2$

$\Rightarrow$ $\sum f_i=50$ [ Given ]

$\therefore$ $33+f_1+f_2=50$

$\therefore$ $f_1=17-f_2$ ------ ( 1 )

$\Rightarrow$ $Mean=\frac{\sum f_i{x}_i}{\sum f_i}$

$78=\frac{2575+65f_1+85f_2}{50}$ $3900=2575+65f_1+85f_2$

$1325=65f_1+85f_2$

$1325=65(17-f_2)+85f_2$

$1325=1105-65f_2+85f_2$

$220=20f_2$

$\therefore$ $f_2=\frac{220}{20}=11$ ----- ( 2 )

Substituting value of ( 2 ) in ( 1 ),

$\therefore$ $f_1=17-11=6$

$\therefore$ $f_1=6$ and $f_2=11$