Question

The mean of the following frequency distributed of $200$ observation is $332$. Find the value of $x$ and $y$

Class	$100-150$	$150-200$	$200-250$	$250-300$	$300-350$	$350-400$	$400-450$	$400-450$	$500-550$
Frequency	$4$	$8$	$x$	$42$	$50$	$y$	$32$	$6$	$4$

Answer 1

$Class$	$Class-Mark$ $x_i$	$f_i$	$f_i{x}_i$
$100-150$	$125$	$4$	$500$
$150-200$	$175$	$8$	$1400$
$200-250$	$225$	$x$	$225x$
$250-300$	$275$	$42$	$11550$
$300-350$	$325$	$50$	$16250$
$350-400$	$375$	$y$	$375y$
$400-450$	$425$	$32$	$13600$
$450-500$	$475$	$6$	$2850$
$500-550$	$525$	$4$	$2100$
		$\sum f_i=146+x+y$	$\sum f_i{x}_i=48250+225x+375y$

$\Rightarrow$ $\sum f_i=200$ [ Given ]

$\Rightarrow$ $sum{f}_i=146+x+y$

$\Rightarrow$ $200=146+x+y$

$\Rightarrow$ $x+y=54$

$\therefore$ $x=54-y$ ---- ( 1 )

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

$\Rightarrow$ $332=\frac{48250+225x+375y}{200}$

$\Rightarrow$ $66400=48250+225(54-y)+375y$

$\Rightarrow$ $18150=12150-225y+375y$

$\Rightarrow$ $6000=150y$

$\therefore$ $y=40$

Substituting $y=40$ in ( 1 ) we get,

$x=14$

$\therefore$ $x=14$ and $y=40$