Question

find the mode of the following frequency distribution class 0 to 10 ten to Twenty 20 to 30 30 to 40 40 to 50 50 to 60 60 to 70 frequency 5 8 15 20 14 8 3​

Answer 1

$\underline{\sf\ \ \ Given:-}$

$\begin{tabular}{|c|c|c|c|c|c|c|c|}\cline{1-8}\it Class\ Interval &\sf 0-10&\sf 10-20&\sf 20-30&\sf 30-40&\sf 40-50&\sf 50-60&\sf 60-70\\\cline{1-8}\it\ frequency (f_i)&\sf 5&\sf 8&\sf 15&\sf 20&\sf 14&\sf 8&\sf 3 \\\cline{1-8} \end{tabular}$

$\underline{\sf\ \ \ To\ Find :-}$

• Mode of the given frequency distribution

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

• Modal class is the class of maximum frequency.

$\rule{280}{1.5}$

$\underline{\boxed{\sf\ Mode= \ell+\bigg\lgroup\dfrac{f_o-f_1}{2f_o-f_1-f_2}\bigg\rgroup\times h}}\\ \\ \bullet\sf\ \ Where :-\\ \\ \\\star\sf\ \ \ell= Lower\ limit\ of\ Modal\ Class\\ \\ \\\star\sf\ \ f_o= frequency\ of\ Modal\ Class\\ \\ \\\star\sf\ \ f_1=frequency\ of\ the\ class\ preceding\ the\ Modal\ class\\ \\ \\ \star\sf\ \ f_2= frequency\ of\ the\ class\ following\ the\ Modal\ class\\ \\ \\\star\sf\ \ h= width\ of\ Modal\ Class$

• Here, the maximum frequency is 20 and the corresponding class is (30-40)

• So ,(30-40) is the Modal class

• So we have -

$\bullet\sf\ \ell= 30\ \ ; \ \ \bullet\sf\ h=(40-30)=10\\ \\ \\ \bullet\sf\ f_o=20\ \ ; \ \bullet\sf\ f_1= 15\ \ ; \ \bullet\sf\ f_2=14$

$\rule{280}{1.5}$

• Now Mode :-

$\dashrightarrow\sf\ Mode= \ell+h\times\bigg\lgroup\dfrac{f_o-f_1}{2f_o-f_1-f_2}\bigg\rgroup\\ \\ \\ \dashrightarrow\sf\ Mode=30+10\times \bigg\lgroup\dfrac{20-15}{2(20)-15-14}\bigg\rgroup\\ \\ \\ \dashrightarrow\sf\ Mode= 30+10\times \bigg\lgroup\dfrac{5}{40-29}\bigg\rgroup\\ \\ \\ \dashrightarrow\sf\ Mode= 30+\bigg\lgroup\dfrac{10\times5}{11}\bigg\rgroup\\ \\ \\ \dashrightarrow\sf\ Mode= 30+\bigg\lgroup\cancel{\dfrac{50}{11}}\bigg\rgroup\\ \\ \\ \dashrightarrow\sf\ Mode= 30+4.54\\ \\ \\ \dashrightarrow\underline{\boxed{\sf\ Mode= 34.54}}$

Answer 2

Answer:

$\begin{tabular}{|c|c|}\cline{1-2}\sf Class Interval&\sf Frequency \\\cline{1-2}\sf0-10&\sf5\\\cline{1-2}\sf10-20&\sf8\\\cline{1-2}\sf20-30&\sf15\\\cline{1-2}\sf30-40&\sf20\\\cline{1-2}\sf40-50&\sf14\\\cline{1-2}\sf50-60&\sf8\\\cline{1-2}\sf60-70&\sf3\\\cline{1-2}\end{tabular}$

⠀

Here maximum frequency is 20, the class corresponding to this 30 - 40

⠀

• So modal class is 30-40
• Now lower limit of modal class ( l ) = 30
• Frequency ( f₁ ) of modal class = 15
• Frequency ( f₀ ) of preceding modal class = 20
• Frequency ( f₂ ) of the class succeeding modal class = 14
• Class size ( h ) = 10

⠀

$\underline{\bigstar\:\textsf{According to the given Question :}}$

$:\implies\sf Mode= l+\bigg[\dfrac{f_o-f_1}{2f_o-f_1-f_2}\bigg]\times h\\\\\\:\implies\sf Mode=30+\bigg[\dfrac{20-15}{2(20)-15-14}\bigg]\times10\\\\\\:\implies\sf Mode= 30+\bigg[\dfrac{5}{40-29}\bigg]\times10\\\\\\:\implies\sf Mode=30+\bigg[\dfrac{10\times5}{11}\bigg]\\\\\\:\implies\sf Mode= 30+\dfrac{50}{11}\\\\\\:\implies\sf Mode= 30+4.54\\\\\\:\implies\underline{\boxed{\sf\ Mode= 34.54}}$

⠀

$\therefore\:\underline{\textsf{Hence, Mode of the given data is \textbf{34.54}}}$