Question

Compute the missing frequencies ${}^{\prime }{f}_1^{\prime }$ and ${}^{\prime }{f}_2^{\prime }$ in the following data, if the mean is $166\frac{9}{26}$ and the sum of the observation is 52.

Classes	Frequency
140-150	5
150-160	$f_1$
160-170	20
170-180	$f_2$
180-190	6
190-200	2
Total	52

• A. $f_1=7,f_2=3$
• B. $f_1=10,f_2=6$
  
• C. $f_1=9,f_2=8$
• D. None of these

Answer 1

The correct option is D None of these

Given:- $\Sigma f=52$

classes	frequency $\left(f_i\right)$	$x_i=\frac{\text{lower limit + upper limit}}{2}$	$x_i{f}_i$
140-150	5	145	725
150-160	$f_1$	155	$155f_1$
160-170	20	165	3300
170-180	$f_2$	175	$175f_2$
180-190	6	185	1110
190-200	2	195	390
	$\Sigma f_i=52$		$\Sigma x_i{f}_i=5525+155f_1+175f_2$

Also $\Sigma f_i=33f_1+f_2=52$

$\Rightarrow f_1+f_2=19⟶eq.\left(i\right)$

Now Mean = $\frac{\Sigma x_i{f}_i}{\Sigma f_i}=\frac{5525+155f_1+175f_2}{52}$

Given:- Mean = $\frac{4325}{26}$

$\Rightarrow \frac{5525+155f_1+175f_2}{52}=\frac{4325}{26}$

$\Rightarrow 5525+155f_1+175f_2=8650$

$\Rightarrow 155f_1+175f_2=8650-5525=3125$

$31f_1+35f_2=625⟶eq.\left(ii\right)$

from eq. (i) $\&$ (ii), we get

$f_2=9\Rightarrow f_1=10$

D) None of these