Question

The mean of the following frequency table is $50$. Find the values of $x$ and $y$.

Class-interval	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$	Total
Frequency	$17$	$x$	$32$	$y$	$19$	$120$

Answer 1

Let the assumed mean be $A=50$ and $h=20$

Table for step deviation is given:

Class Interval	Frequency, $f_i$	Mid Value	$u_i=\frac{x_i-A}{h}$	$f_i{u}_i$
$0-20$	$17$	$10$	$-2$	$-34$
$20-40$	$x$	$30$	$-1$	$-x$
$40-60$	$32$	$50$	$0$	$0$
$60-80$	$y$	$70$	$1$	$y$
$80-100$	$19$	$90$	$2$	$38$
Total	$68+x+y$			$4-x+y$

We have $N=\sum f_i=120$

$\Rightarrow 68+x+y=120$

$\Rightarrow x+y=52$ ... (i)

Mean $=50$

Mean $=A+(\frac{1}{N}\sum f_i{u}_i)h$

$\Rightarrow 50=50+(\frac{1}{120}\times (4-x+y))\times 20$

$\Rightarrow x-y=4$ ... (ii)

From (i) and (ii), we get

$x=28,y=24$