Question

Answer the following :
The median of 230 observation is 46. Find a and b.

Class	10-20	20-30	30-40	40-50	50-60	60-70	80-90
Frequency	12	30	a	65	b	25	18

Answer 1

First finding Cumulative Frequencies(cf) of all classes

cf of class $10$ to $20=12$

cf of class $20$ to $30=12+30=42$

cf of class $30$ to $40=42+a$

cf of class $40$ to $50=107+a$

cf of class $50$ to $60=107+a+b$

cf of class $60$ to $70=132+a+b$

cf of class $70$ to $80=150+a+b$

As total number of observations are, $n=230$ so $230=150+a+b$, because cf of last interval is equal to $n$.

$a+b=80$ ... $\left(1\right)$

Median$=46$ that is between $40$ and $50$, hence median class$=40-50$

$f=$ frequency of median class $=65$

$cf=$ cumulative frequency $=42+a$ ... (cf of previous class)

$l=$ lower limit of median class $=40$

$h=50-40=10$

Median$=l+\frac{\frac{n}{2}-cf}{f}\times h$

$46=40+\frac{\frac{230}{2}-(42+a)}{65}\times 10$

$\frac{6}{10}=\frac{73+a}{65}$

$a=34$

From $\left(1\right)$:

$34+b=80$

$b=46$

Hence $a=34$ and $b=46$