Question

find the mode of given data
weight in kg- 30-40, 40-50, 50-60, 60-70, 70-80
No. of students- 11,29,6,3,1​

Answer 1

Given data is:

$\begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 30 - 40 & 11  \\40 - 50 & 29 \\50 - 60 & 6 \\60 - 70 & 3 \\70 - 80 & 1 \end{array}$

Formula for mode:

$\boxed{ \boxed{\sf{mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }}}$

Where,

l is lower limit of modal class.

$\sf{f_1}$ is frequency of modal class

$\sf{f_0}$ is frequency of class preceding modal class

$\sf{f_2}$ is frequency of class succeeding modal class, and

h is class height.

Solution:

Here, the modal class is 40-50 so,

$\sf{l = 40}$

$\sf{h = 10}$

$\sf{f_0 = 11}$

$\sf{f_1 = 29}$

$\sf{f_2 = 6}$

Using the formula for mode

$\implies\sf{mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }$

$\implies\sf{mode = 40 + \bigg(\dfrac{29 - 11}{2(29) - 11 - 6} \bigg) \times 10 }$

$\implies\sf{mode = 40 + \bigg(\dfrac{18}{41} \bigg) \times 10 }$

$\boxed{ \boxed{ \implies\sf{mode = 44.39}}}$

Answer 2

$\Large{\bf{\green{\underline{GiVeN\:DaTa\::}}}}$

$\boxed{\begin{array}{c|c|c|c|c|c}\bf Weight\:(in\:kg) & \sf 30-40 & \sf 40-50 & \sf 50-60 &\sf 60-70 & \sf 70-80 \\ \frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \quad\qquad}{}&\frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}\\ \bf No. \:of \:students & \sf 11 & \sf 29 & \sf 6 & \sf 3 & \sf 1\end{array}}$

$\\ \Large{\bf{\pink{\underline{To\:FiNd\::}}}}$

• Mode.

$\\ \Large{\bf{\purple{\underline{CaLcUlAtIoN\::}}}}$

➣ The mode is that value in a series of observation which occurs with greatest frequency.

➻ For calculating the mode we take the given data and find the modal class where the frequency is highest. Then we use the following formula as,

$\red\bigstar\:\:{\underline{\blue{\boxed{\bf{\green{Mode\:=\:L\:+\:\bigg(\:\dfrac{f_1\:-\:f_o}{2f_1\:-\:f_o\:-\:f_2}\:\bigg)\:h\:}}}}}}$

$\bf\pink{Where,}$

• L is lower interval of mode class.

• fₒ is the frequency of proceeding modal class.

• f₁ is the frequency of modal class.

• f₂ is the frequency of succeeding modal class.

• h is height of class interval.

➛ Here, we can see that the highest frequency is at '40-50'.

➛ So, we can say that class 40-50 is the modal class.

$\bf\red{Thus,}$

• L = 40

• fₒ = 11

• f₁ = 29

• f₂ = 6

• h = 10

$:\implies\:\:\bf{Mode\:=\:40\:+\:\bigg(\:\dfrac{29\:-\:11}{2\times{29}\:-\:11\:-\:6}\:\bigg)\times{10}\:}$

$:\implies\:\:\bf{Mode\:=\:40\:+\:\bigg(\:\dfrac{18}{58\:-\:17}\:\bigg)\times{10}\:}$

$:\implies\:\:\bf{Mode\:=\:40\:+\:\bigg(\:\dfrac{18}{41}\times{10}\:\bigg)}$

$:\implies\:\:\bf{Mode\:=\:40\:+\:\dfrac{180}{41}\:}$

$:\implies\:\:\bf{Mode\:=\:40\:+\:4.39\:}$

$:\implies\:\:\bf\orange{Mode\:=\:44.39\:}$

━─━─━─━─━─━─━─━─━─━─━─━─━─━─━

$\Large\bf\blue{Therefore,}$

The mode of the given data is 44.39.