Question

The median of $230$ observations of the following frequency distribution is $46$. Find $a$ and $b$:

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

Answer 1

We will add one more column of cumulative frequency as follows:

Class	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Frequency	$12$	$30$	$a$	$65$	$b$	$25$	$18$
cumulative frequency	$12$	$42$	$42+a$	$107+a$	$107+a+b$	$132+a+b$	$150+a+b$

Now the total number of observations will be equal to the cumulative frequency.

$\therefore$ $150+a+b=230$

Now we will use the median formula to form the second equation, as follows:

Median is $l+h\left(\frac{n/2-cf}{f}\right)$

$h=10$

$l=40$

$cf=42+a$

$n/2=230/2=115$

$f=65$

Now putting these values, we get:

Median$=46=40+10\left(\frac{115-(42+a)}{65}\right)$

$\Rightarrow a=34$

Putting this value of $a$ in equation $1$ , we get:

$b=80-a$

$b=46$

Thus the value of $a$ is $34$ and $b$ is $46$.