Question

find the mode of the frequency distribution table given below class interval 0-10, 10-20, 20-30, 30-40, 40-50 frequency 7, 9, 15, 11, 8​

Answer 1

Given data is

$\begin{gathered} \begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 0 - 10 & 7  \\10 - 20 & 9 \\20 - 30 & 15 \\30 - 40 & 11 \\40 - 50 & 8 \end{array}\end{gathered}$

We know,

Mode is given by

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

Where,

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

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

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

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

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

Now,

Here,

Modal class = 20 - 30

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

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

$\: \: \: \: \: \: \: \: \bull \: \: \: \: \: \sf \: f_0 = 9$

$\: \: \: \: \: \: \: \: \bull \: \: \: \: \: \sf \: f_1 = 15$

$\: \: \: \: \: \: \: \: \bull \: \: \: \: \: \sf \: f_2 = 11$

On substituting all these values in above formula,

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

$\rm :\longmapsto\:{{\bf{Mode = 20+ \bigg(\dfrac{15 - 9}{2 \times 15 - 9 - 11} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20+ \bigg(\dfrac{6}{30 -20} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20+ \bigg(\dfrac{6}{10} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 20+ 6 }}}$

$\rm :\longmapsto\:{{\bf{Mode = 26 }}}$

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 :-

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