Question

Find the mode of the following
distribution:
Cl 0-10 10-20 20-30 30-40 40-50
frequency
3
5
9
5
3
گے​

Answer 1

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

$\begin{gathered} \begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 0 - 10 & 3  \\10 - 20 & 5 \\20 - 30 & 9 \\30 - 40 & 5 \\40 - 50 & 3 \end{array}\end{gathered}$

Formula of Mode,

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

Where,

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

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

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

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

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

So,

Here,

Modal class is 20 - 30

It implies,

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

$\: \: \: \: \: \: \: \bull \: \: \sf \:l \: = 20$

$\: \: \: \: \: \: \: \bull \: \: \sf \:f_0 \: = 5$

$\: \: \: \: \: \: \: \bull \: \: \sf \:f_1 \: = 9$

$\: \: \: \: \: \: \: \bull \: \: \sf \:f_2 \: = 5$

Hence,

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

$\rm :\longmapsto\:{{\bf{Mode = 20 + \bigg(\dfrac{9 - 5}{2 \times 9 - 5 -5} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20 + \bigg(\dfrac{4}{18 - 10} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20 + \bigg(\dfrac{4}{8} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20 + \bigg(\dfrac{1}{2} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20 + 5}}}$

$\rm :\longmapsto\:{{\bf{Mode = 25}}}$

Additional Information :-

1. Mean using Direct Method

$\dashrightarrow\sf Mean = \dfrac{ \sum f_i x_i}{ \sum f_i}$

2. Mean using Short Cut Method

$\dashrightarrow\sf Mean =A + \dfrac{ \sum f_i d_i}{ \sum f_i}$

3. Mean using Step Deviation Method

$\dashrightarrow\sf Mean =A + \dfrac{ \sum f_i u_i}{ \sum f_i} \times h$

4. Median

$\dashrightarrow\sf M= l + \Bigg \{h \times \dfrac{ \bigg( \dfrac{N}{2} - cf \bigg)}{f} \Bigg \}$