Question

The mean of the following data is $42$. Find the missing frequencies $x$ and $y$ if the sum of frequencies is $100$.

Class interval	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	7	10	x	13	y	10	14	9

Answer 1

Given that mean of data is $42$

sum of frequencies (N) or $\sum f=100$

Class interval $0-10$ $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$

Frequency $7$ $10$ x $13$ y $10$ $14$ $9$

Class interval frequency($f_i$) mid value of $\left(x_i\right)$ $f_i{x}_i$

$0-10$ $7$ $5$ $35$

$10-20$ $10$ $15$ $150$

$20-30$ x $25$ $25x$

$30-40$ $13$ $35$ $455$

$40-50$ y $45$ $45y$

$50-60$ $10$ $55$ $550$

$60-70$ $14$ $65$ $910$

$70-80$ $9$ $75$ $675$

___________ ______________

$63+x+y$ $2775+25x+45y$

Given $N=100$

$\Rightarrow 63+x+y=100$

$x+y=100-63=37$

$x+y=37$ ,,,,,,,,,$\left(1\right)$

we know mean $=\sum f_i{x}_i/\sum f_i$

$42=\frac{2775+25x+45y}{100}$

$4200=2775+25x+145y$

$\Rightarrow 25x+45y=1425$

$5(5x+9y)=5\times 285$

$5x+9y=285$ .............$\left(2\right)$

$\left(1\right)\times 9=9x+9y=333$

$\left(2\right)=5x+9y=285$

___________________

$4x=48$

$\Rightarrow x=12$

$\left(1\right)$ $\Rightarrow x+y=37$

$\Rightarrow y=37-x=37+12=25$

$y=25$.