Question

Find the mode of the following frequency distribution:

Class interval: $25-30,30-35,35-40,40-45,45-50,50-55$

Frequency: $25,34,50,42,38,14$

Answer 1

Find the mode of the given data

Given frequency distribution is:

Class interval	$25-30$	$30-35$	$35-40$	$40-45$	$45-50$	$50-55$
Frequency	$25$	$34$	$50$	$42$	$38$	$14$

For this continuous data set, class interval with highest frequency is $35-40$ with a frequency of $50$.

We know that, $\mathrm{mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h$, where

$l=35$, lower limit of modal class

$f_1=50$, frequency of the modal class

$f_0=34$ , frequency of the class preceding modal class

$f_2=42$ , frequency of the class succeeding modal class

$h=5$ , size of the class interval

So, the mode for modal class $35-40$ will be,

$$\begin{gathered} \Rightarrow \mathrm{mode}=35+\left\{\frac{50-34}{(2\times 50)-34-42}\right\}\times 5 \\ \Rightarrow \mathrm{mode}=35+\left(\frac{16}{100-76}\right)\times 5 \\ \Rightarrow \mathrm{mode}=35+\frac{16\times 5}{24} \\ \Rightarrow \mathrm{mode}=35+\frac{10}{3} \\ \Rightarrow \mathrm{mode}=\frac{115}{3}=38.33 \end{gathered}$$

Hence, the mode of the given data is $38.33$.