The following is an incomplete distribution:

Variable	Frequency
10-20	12
20-30	30
30-40	$f_1$
40-50	65
50-60	$f_2$
60-70	25
70-80	18

 

If the median value is 46 and there are 230 items, Find the missing frequencies $f_1$ and $f_2$.

For continuous data,  $\mathrm{Median}=l+\frac{\left(\frac{N}{2}-\mathrm{cf}\right)}{{\mathrm{f}}_{\mathrm{m}}}\times \mathrm{c}$

Where N = Total cumulative frequency

$l =$ the lower limit of the median class Interval; 

Median Class = the class interval having a cumulative frequency $\ge \frac{\mathrm{N}}{2}$ 

$cf=$cumulative frequency of the class interval just before the median

$\mathrm{f}=$ frequency of the median class interval;

$c=$width of the median class interval

Here given $N=230$ and from table $N=f_1+f_2+150$

Given median = 46 ,

this cumulative frequency lies within interval $(40-50)$;

$\frac{\mathrm{N}}{2}=115 , l=40 , cf={\mathrm{f}}_1+42 , \mathrm{f}=65 , c=10$

Therefore $40+\left(\frac{115-\left({\mathrm{f}}_1+42\right)}{65}\times 10\right)=46$

$\Rightarrow \left(\frac{115-42-{\mathrm{f}}_1}{65}\times 10\right)=46-40$

$\Rightarrow \left(73-{\mathrm{f}}_1\right)\times 10=6\times 65$

$\Rightarrow 730-10{\mathrm{f}}_1=390$

$\Rightarrow -10{\mathrm{f}}_1=390-730$

$\Rightarrow 10f1=340$

$\Rightarrow f_1=34$

Now

${\mathrm{f}}_1+{\mathrm{f}}_2+150=230$

$\Rightarrow {\mathrm{f}}_2=230-150-34$

$\Rightarrow {\mathrm{f}}_2=46$