Question

If the median of the following data is $56\frac{2}{3}$, then the value of x is

Class Interval	40-45	45-50	50-55	55-60	60-65	65-70	70-75
Frequency	2	3	x	6	6	3	2

• A. 12
• B. 8
• C. 7
• D. 10

Answer 1

The correct option is B 8

Class Interval	Frequency	cf
40-45	2	2
45-50	3	5
50-55	x	5+x
55-60	6	11+x
60-65	6	17+x
65-70	3	20+x
70-75	2	22+x

Here the total number of observatives (N) is 22 + x

$\therefore \frac{N}{2}=\frac{22+x}{2}$

$Median=56\frac{2}{3}$
So, the median lies in the interval
55-60.
Thus, 55 - 60 is the median class.
Here,
Lower limit, (l) = 55
Cumulative frequency of class preceding the median class, (cf) = 5 + x
Frequency of the median class (f) = 6
Class size (h) = 5
$\therefore Median=l+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$

$\Rightarrow \frac{170}{3}=55+\left[\frac{\frac{22+x}{2}-(5+x)}{12}\right]\times 5$

$\Rightarrow \frac{170}{3}-55=\left[\frac{22+x-2(5+x)}{12}\right]\times 5$

$\Rightarrow \frac{170-165}{3}=\left(\frac{22+x-10-2x}{12}\right)\times 5$

$\Rightarrow \frac{5}{3}=\left(\frac{12-x}{12}\right)\times 5$

$\Rightarrow \frac{5}{3}\times \frac{12}{5}=12-x$

$\Rightarrow 4=12-x$
$\Rightarrow x=12-4$
$\Rightarrow x=8$

Hence, the correct answer is option b.