Question

Find the mean, median and mode of the following data:

Classes:	$0-50$	$50-100$	$100-150$	$150-200$	$200-250$	$250-300$	$300-350$
Frequency:	$2$	$3$	$5$	$6$	$5$	$3$	$1$

Answer 1

Class interval	Mid value $x_i$	Frequency $f_i$	$f_i{x}_i$	$cf$
$0-50$	$35$	$2$	$50$	$2$
$50-100$	$75$	$3$	$225$	$5$
$100-150$	$125$	$5$	$625$	$10$
$150-200$	$175$	$6$	$1050$	$16$
$200-250$	$225$	$5$	$1127$	$21$
$250-300$	$275$	$3$	$825$	$24$
$300-350$	$325$	$1$	$325$	$25$
		$\sum f_i=25$	$\sum f_i{x}_i=4225$

$\Rightarrow$ $Mean=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{4225}{25}=169$

$\Rightarrow$ We have $N=25$$.Then,$\dfrac{N}{2}=12.5$

$\Rightarrow$ So, median class is $150-200$.

$l=$ lower limit of the modal class

$h=$ size of the class intervals

$f=$ frequency of the modal class

$f_1=$ frequency of the class preceding the modal class

$f_2=$ frequency of the class succeed in the modal class.

$\therefore$ $l=150,h=200-150=50,f=6,cf=10$

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

$\Rightarrow$ $Median=150+\frac{12.5-10}{6}\times 50$

$\Rightarrow$ $Median=150+\frac{125}{6}$

$\therefore$ $Median=150+20.83=170.83$.

$\Rightarrow$ Here maximum frequency is $6$, then the corresponding class $150-200$ is the modal class.

$\Rightarrow$ $l=150,h=50,f=6,f_1=5f_2=5$

$\Rightarrow$ $Mode=l+\frac{f-f_1}{2f-f_1-f_2}\times h$

$\Rightarrow$ $Mode=150+\frac{6-5}{2\times 6-5-5}\times 50$

$\Rightarrow$ $Mode=150+\frac{50}{2}$

$\therefore$ $Mode=150+25=175$