Question

The mean of the following distribution is $50$. Find a.

x	$10$	$30$	$50$	$70$	$90$
y	$17$	$5a+3$	$32$	$7a-11$	$18$

Answer 1

Mean $=\frac{{\sum }_{i=1}^5{f}_i{x}_i}{{\sum }_{i=1}^5{f}_i}$

So, first ${\sum }_{x=1}^5{f}_1{x}_1=(10\times 17)+\left(30(5a+3)\right)+(50\times 32)+70\times (7a-11)+(90\times 19)$

$=170+150a+90+1600+490a-770+1710$

$=2800+640a$

Now, ${\sum }_{i=1}^5{f}_i=17+5a+3+32+7a-11+19$

$=60+12a$

Mean $=\frac{{\sum }_{i=1}^5{f}_i{x}_i}{{\sum }_{i=1}^5{f}_i}$

$\Rightarrow 50=\frac{2800+640a}{60+12a}$

$\Rightarrow 50(60+12a)=2800+640a$

$\Rightarrow 3000-2800=640a-600a$

$a=5$.