Question

The mode for the following frequency distribution is:

Size of items	Frequency	Size of items	Frequency
0-4	5	20-24	10
4-8	7	24-28	6
8-12	9	28-32	3
12-16	17	32-36	1
16-20	12	36-40	0

• A. $32.66$
• B. $14.46$
• C. $14$
• D. none of these

Answer 1

The correct option is B $14.46$

For finding the mode of a grouped frequency distribution, we need to identify the modal class first.

Here, the maximum frequency is $17$ and the corresponding class is $12-16$.

Hence, $12-16$ is the modal class.
We know that,

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

Where, $l\to$ lower limit of the modal class

$f\to$ frequency of the modal class

$f_1\to$ frequency of the class preceding the modal class

$f_2\to$ frequency of the class succeeding the modal class

$h\to$ class width

Here, $l=12,h=4,f=17,f_1=9$ and $f_2=12$

$=12+\frac{17-9}{34-9-12}\times 4$
$\Rightarrow$ Mode $=12+\frac{8}{13}\times 4=12+\frac{32}{13}=12+2.46=14.46$

Hence, the mode of the give frequency distribution is $14.46$.