find the mode of given data

class:- 1500-2000 2000-2500 2500-3000 3000-3500 3500-4000 4000-4500 4500-5000
frequenc

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1. [tex]\large\underline{\sf{Solution-}}[/tex]

   Given data is

   [tex]\begin{gathered} \begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 1500 – 2000 & 14 \\2000 – 2500 & 56 \\2500 – 3000 & 60 \\3000 – 3500 & 86 \\3500 – 4000 & 74\\4000 – 4500 & 62\\4500 – 5000 & 48 \end{array}\end{gathered}[/tex]

   We know,

   Mode is evaluated as

   [tex]\boxed{ \boxed{\sf{Mode = l + \bigg(\dfrac{f_1 – f_0}{2f_1 – f_0 – f_2} \bigg) \times h }}}[/tex]

   where,

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: l \: is \: lower \: limit \: of \: modal \: class[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_0 \: is \: frequency \: of \: class \: preceeding \: modal \: class[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_1 \: is \: frequency \: of \: modal \: class[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_2 \: is \: frequency \: of \: class \: succeeding \: modal \: class[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: h \: is \: height \: of \: modal \: class[/tex]

   Here,

   Modal class = 3000 – 3500

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: l = 3000[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: h = 500[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_0 = 60[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_1 = 86[/tex]

   [tex] \: \: \: \: \: \: \: \: \bull \rm \: f_2 = 74[/tex]

   On substituting all these values in formula of mode, we have

   [tex]\rm :\longmapsto\:{{\bf{Mode = l + \bigg(\dfrac{f_1 – f_0}{2f_1 – f_0 – f_2} \bigg) \times h }}}[/tex]

   [tex]\rm :\longmapsto\:{{\sf{Mode = 3000 + \bigg(\dfrac{86 – 60}{2 \times 86 – 60 – 74} \bigg) \times 500 }}}[/tex]

   [tex]\rm :\longmapsto\:{{\sf{Mode = 3000 + \bigg(\dfrac{26}{172 – 134} \bigg) \times 500 }}}[/tex]

   [tex]\rm :\longmapsto\:{{\sf{Mode = 3000 + \bigg(\dfrac{26}{38} \bigg) \times 500 }}}[/tex]

   [tex]\rm :\longmapsto\:{{\sf{Mode = 3000 + 342.105 }}}[/tex]

   [tex]\rm :\longmapsto\:{{\sf{Mode = 3342.105 }}}[/tex]

   Additional Information :–

   Mean using Direct Method :-

   [tex]\dashrightarrow\sf Mean = \dfrac{ \sum f_i x_i}{ \sum f_i}[/tex]

   Mean using Short Cut Method

   [tex]\dashrightarrow\sf Mean =A + \dfrac{ \sum f_i d_i}{ \sum f_i}[/tex]

   Mean using Step Deviation Method

   [tex]\dashrightarrow\sf Mean =A + \dfrac{ \sum f_i u_i}{ \sum f_i} \times h[/tex]

   Median

   [tex]\dashrightarrow\sf M= l + \Bigg \{h \times \dfrac{ \bigg( \dfrac{N}{2} – cf \bigg)}{f} \Bigg \}[/tex]

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