Question

28. Find the mode of the following frequency distribution.
CI
0-9 10-19 20-29 30-39 40-49 50-59 60-69
f 4. 7 14 20 13 7 5
​

Answer 1

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

$\begin{gathered} \begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 0 - 9 & 4  \\10 - 19 & 7 \\20 - 29 & 14 \\30 - 39 & 20 \\40 - 49 & 13\\50 - 59 & 7\\60 - 69 & 5 \end{array}\end{gathered}$

Here, the given class intervals are not in inclusive form. So we first convert in to inclusive form by Subtracting 0.5 in lower limit and adding 0.5 in upper limit of each class.

The given frequency distribution in inclusive form is given below :-

$\begin{gathered} \begin{array}{|c|c|} \bf{x_i} & \bf{f_i} \\ 0 - 9.5 & 4  \\9.5 - 19.5 & 7 \\19.5 - 29.5 & 14 \\29.5 - 39.5 & 20 \\39.5 - 49.5 & 13\\49.5 - 59.5 & 7\\60.5 - 69.5 & 5 \end{array}\end{gathered}$

We know,

Formula for mode

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

Where,

l is lower limit of modal class.

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

$\sf{f_1} \: is \: frequency \: of \: modal \: class</p><p>$

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

and

h is the height of modal class

Here,

Modal class = 29.5 - 39.5

So,

$\rm :\longmapsto\:l \: = \: 29.5$

$\rm :\longmapsto\:f_0 = 14$

$\rm :\longmapsto\:f_1 = 20$

$\rm :\longmapsto\:f_2 = 13$

$\rm :\longmapsto\:h = 10$

So,

On substituting all these values in above formula,

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

$\rm :\longmapsto\:{{\sf{Mode = 29.5 + \bigg(\dfrac{20 - 14}{2 \times 20 - 14 - 13} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\sf{Mode = 29.5 + \bigg(\dfrac{6}{40 - 27} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\sf{Mode = 29.5 + \bigg(\dfrac{6}{13} \bigg) \times 10 }}}$

$\rm :\longmapsto\:{{\sf{Mode = 29.5 + \bigg(\dfrac{60}{13} \bigg)}}}$

$\rm :\longmapsto\:{{\sf{Mode = 29.5 + \bigg(4.615 \bigg)}}}$

$\rm :\longmapsto\:{{\sf{Mode = 34.115}}}$

Additional Information :-

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

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

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

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