Question

Find the mode of the following distribution.
(i)

Class-interval:	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency:	$5$	$8$	$7$	$12$	$28$	$20$	$10$	$10$

(ii)

Class-interval:	10-15	15-20	20-25	25-30	30-35	35-40
Frequency:	$30$	$45$	$75$	$35$	$25$	$15$

(iii)

Class-interval:	25-30	30-35	35-40	40-45	45-50	50-60
Frequency:	$25$	$34$	$50$	$42$	$38$	$14$

Answer 1

We know,

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

$l=$ 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.

$\left(i\right)$ Here, the maximum frequency is $28$ then the corresponding class $40-52$ is modal class.

$\Rightarrow$ $l=40,h=50-40=10,f=28,f_1=12,f_2=20$

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

$\Rightarrow$ $Mode=40+\frac{28-12}{2\times 28-12-20}\times 10$

$\Rightarrow$ $Mode=40+\frac{160}{24}$

$\Rightarrow$ $Mode=40+6.67=46.67$

$\left(ii\right)$ Here, the maximum frequency is $75$, then the corresponding class $20-25$ is the modal class.

$\Rightarrow$ $l=20,h=25-20=5,f=75,f_1=45,f_2=35$

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

$\Rightarrow$ $Mode=20+\frac{75-45}{2\times 75-45-35}\times 5$

$\Rightarrow$ $Mode=20+\frac{150}{70}$

$\Rightarrow$ $Mode=20+2.14=22.14$

$\left(iii\right)$ Here, the maximum frequency is $50$ then the corresponding class $35-40$ is the modal class.

$\Rightarrow$ $l=35,h=40-35=5,f=50,f_1=34,f_2=42$

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

$\Rightarrow$ $Mode=35+\frac{50-34}{2\times 50-34-42}\times 5$

$\Rightarrow$ $Mode=35+\frac{80}{24}$

$\Rightarrow$ $Mode=35+3.33=38.33$