Question

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

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \small\boxed{\begin{array}{c |c} \tt{Classes} & \tt{Frequency} \\ \dfrac{\qquad\qquad}{ \sf 0-10} &\dfrac{\qquad\qquad}{ \sf 5} & \\ \dfrac{\qquad\qquad}{ \sf 10-20} &\dfrac{\qquad\qquad}{ \sf 10} & \\ \dfrac{\qquad\qquad}{ \sf 20-30} &\dfrac{\qquad\qquad}{ \sf 18} & \\ \dfrac{\qquad\qquad}{ \sf 30-40} &\dfrac{\qquad\qquad}{ \sf 30} & \\ \dfrac{\qquad\qquad}{ \sf 40-50} &\dfrac{\qquad\qquad}{ \sf 20} & \\ \dfrac{\qquad\qquad}{ \sf 50-60} &\dfrac{\qquad\qquad}{ \sf 12} & \\ \dfrac{\qquad\qquad}{ \sf 60-70} &\dfrac{\qquad\qquad}{ \sf 5} &\end{array}} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

Answer 1

$\large\underline{\sf{Solution-}}$

Calculation of Mode :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \small\boxed{\begin{array}{c |c} \tt{Classes} & \tt{Frequency} \\ \dfrac{\qquad\qquad}{ \sf 0-10} &\dfrac{\qquad\qquad}{ \sf 5} & \\ \dfrac{\qquad\qquad}{ \sf 10-20} &\dfrac{\qquad\qquad}{ \sf 10} & \\ \dfrac{\qquad\qquad}{ \sf 20-30} &\dfrac{\qquad\qquad}{ \sf 18} & \\ \dfrac{\qquad\qquad}{ \sf 30-40} &\dfrac{\qquad\qquad}{ \sf 30} & \\ \dfrac{\qquad\qquad}{ \sf 40-50} &\dfrac{\qquad\qquad}{ \sf 20} & \\ \dfrac{\qquad\qquad}{ \sf 50-60} &\dfrac{\qquad\qquad}{ \sf 12} & \\ \dfrac{\qquad\qquad}{ \sf 60-70} &\dfrac{\qquad\qquad}{ \sf 5} &\end{array}} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

We know that,

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

where,

$\sf \: \: \: \: \: \bull \: \: \: l \: is \: lower \: limit \: of \: modal \: class$

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

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

$\sf \: \: \: \: \: \bull \: \: \: \sf{f_2} \:  is \: frequency \: of \: class \: succeeding \: modal \: class$

$\sf \: \: \: \: \: \bull \: \: \: h \: is \: height \: modal \: class$

Here,

Modal class = 30 - 40

Thus,

$\sf \: \: \: \: \: \bull \: \: \: l = 30$

$\sf \: \: \: \: \: \bull \: \: \: h = 10$

$\sf \: \: \: \: \: \bull \: \: \: f_0 = 18$

$\sf \: \: \: \: \: \bull \: \: \: f_1 = 30$

$\sf \: \: \: \: \: \bull \: \: \: f_2 = 12$

Thus,

$\red{\rm :\longmapsto\:{{\bf{Mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }}}}$

$\rm :\longmapsto\:{{\bf{Mode = 30+ \bigg(\dfrac{30 - 18}{2 \times 30 - 18 - 12} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 30+ \bigg(\dfrac{12}{60 - 30} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 30+ \bigg(\dfrac{12}{30} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 30+ \bigg(\dfrac{12}{3} \bigg) }}}$

$\rm :\longmapsto\:{{\bf{Mode = 30+ 4 }}}$

$\red{\rm :\longmapsto\:{{\bf{Mode = 34 }}}}$

Calculation of Median :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \small\boxed{\begin{array}{c |c|c} \tt{Classes} & \tt{Frequency}& \tt{c.f.} \\ \dfrac{\qquad\qquad}{ \sf 0-10} &\dfrac{\qquad\qquad}{ \sf 5} &\dfrac{\qquad\qquad}{ \sf 5}& \\ \dfrac{\qquad\qquad}{ \sf 10-20} &\dfrac{\qquad\qquad}{ \sf 10}&\dfrac{\qquad\qquad}{ \sf 15} & \\ \dfrac{\qquad\qquad}{ \sf 20-30} &\dfrac{\qquad\qquad}{ \sf 18}&\dfrac{\qquad\qquad}{ \sf 33} & \\ \dfrac{\qquad\qquad}{ \sf 30-40} &\dfrac{\qquad\qquad}{ \sf 30}&\dfrac{\qquad\qquad}{ \sf 63} & \\ \dfrac{\qquad\qquad}{ \sf 40-50} &\dfrac{\qquad\qquad}{ \sf 20}&\dfrac{\qquad\qquad}{ \sf 83} & \\ \dfrac{\qquad\qquad}{ \sf 50-60} &\dfrac{\qquad\qquad}{ \sf 12}&\dfrac{\qquad\qquad}{ \sf 95} & \\ \dfrac{\qquad\qquad}{ \sf 60-70} &\dfrac{\qquad\qquad}{ \sf 5}&\dfrac{\qquad\qquad}{ \sf 100} &\end{array}} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

$\boxed{ \bf Median= l + \Bigg \{h \times \dfrac{ \bigg( \dfrac{N}{2} - cf \bigg)}{f} \Bigg \}}$

Here,

• l denotes lower limit of median class

• h denotes width of median class

• f denotes frequency of median class

• cf denotes cumulative frequency of the class preceding the median class

• N denotes sum of frequency

According to the question,

Here,

• N = 100

So,

• N/2 = 50

Hence,

• Median class = 30 - 40

so,

• l = 30

• h = 10,

• f = 30,

• cf = cf of preceding class = 33

On substituting the values, we get

$\rm :\longmapsto\:{ \bf Median= 30 + \Bigg \{10 \times \dfrac{ \bigg( 50 - 33 \bigg)}{30} \Bigg \}}$

$\rm :\longmapsto\:{ \bf Median= 30 + \Bigg \{ \dfrac{ 17}{3} \Bigg \}}$

$\rm :\longmapsto\:{ \bf Median= \dfrac{90 + 17}{3} }$

$\rm :\longmapsto\:{ \bf Median= \dfrac{107}{3} }$

Calculation of Mean

We know,

Empirical Formula,

$\bf :\longmapsto\:Mode = 3Median - 2Mean$

$\rm :\longmapsto\:34 = 3 \times \dfrac{107}{3} - 2Mean$

$\rm :\longmapsto\:34 = 107 - 2Mean$

$\rm :\longmapsto\:34 - 107 = - 2Mean$

$\rm :\longmapsto\: - 73 = - 2Mean$

$\bf\implies \:Mean = 36.5$