Question

Find the missing frequencies in the following frequency distribution table if N = 100 and Median is 32​

Answer 1

• ${\boxed{\bf{Frequency \: distribution \: table}}}$

$\bf{\underline{\orange{Class\:interval \qquad \qquad \qquad Frequency}}}$

$\bf{\underline{\quad 0 -10 \qquad \qquad \qquad \quad \qquad \qquad \quad 10\quad}}$

$\bf{\underline{\quad 10 -20 \qquad \qquad \qquad \quad \qquad \qquad \quad f_1\quad}}$

$\bf{\underline{\quad 20 -30 \qquad \qquad \qquad \quad \qquad \qquad \quad 25\quad}}$

$\bf{\underline{\quad 30 -40 \qquad \qquad \qquad \quad \qquad \qquad \quad 30\quad}}$

$\bf{\underline{\quad 40 -50 \qquad \qquad \qquad \quad \qquad \qquad \quad f_2\quad}}$

$\bf{\underline{\quad 50-60 \qquad \qquad \qquad \quad \qquad \qquad \quad 10\quad}}$

To find

• Missing frequencies

Solution

$\large{\red{\underline{\boxed{\bf{\red{Median=\left(l + \dfrac{\dfrac{N}{2} - cf}{f}\right)\times{h}}}}}}}$

• $\sf{l = Lower\:limit}$
• $\sf{h =Width\:of\:class}$
• $\sf{f = Frequency}$
• $\sf{cf = Cumulative\:frequency (preceding\;class)}$
• $\sf{M_e=Median}$

$\tt{\underline{\purple{According\:to\:given\: condition}}}$

$\tt{\blue{Let\;the\;missing\;frequencies\;be\;f_1\;and\;f_2}}$

• $\sf{10-20(class\:interval)=f_1}$
• $\sf{40-50(class\:interval=f_2}$
• $\sf{Total\: frequency (N)=100}$

$\implies\tt{10+f_1+25+30+f_2+10=100}$

$\implies\tt{75+f_1+f_2=100}$

$\implies\tt{f_1+f_2=100-75}$

$\implies\tt{f_1+f_2=25}$

• $\sf{l = 30}$
• $\sf{h =10}$
• $\sf{f = 30}$
• $\sf{cf = (10+f_1+25)=f_1+35}$
• $\sf{M_e=32}$

$\implies\tt{32=30 +\left( \dfrac{ \dfrac{100}{2} - (f_1+35)}{30}\right)\times{10}}$

$\implies\tt{32-30=\left( \dfrac{50 - (f_1+35)}{30}\right)\times{10}}$

$\implies\tt{2=\left( \dfrac{50 - f_1 - 35}{30}\right)\times{10}}$

$\implies\tt{2=\left( \dfrac{50 - 35 - f_1}{30}\times{10}\right)}$

$\implies\tt{2=\left( \dfrac{15 - f_1}{3}\right)}$

$\implies\tt{2\times{3}}=15 - f_1$

$\implies\tt{6=15-f_1}$

$\implies\tt{15-6=f_1}$

$\implies\tt{f_1=9}$

• $\tt\underline{\purple{Now\:put\:the\:value\:of\:f_1}}$

$\implies\tt{f_1+f_2=25}$

$\implies\tt{9+f_2=25}$

$\implies\tt{f_2=25-9}$

$\implies\tt{f_2=16}$

• $\underline{\tt{\purple{Hence,missing\: frequencies}}}$

$\tt{\qquad \qquad \qquad f_1=9}$

$\tt{\qquad \qquad \qquad f_2=16}$

Answer 2

AnsWer :

$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency&\sf Cummulative \: Frequency\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0-10&\sf 10&\sf 10 \\\\\sf 10-20 &\sf x&\sf 10 + x \\\\\sf 20-30 &\sf 25 &\sf 35 + x \\\\\sf 30-40&\sf 30&\sf 65 + x\\\\\sf 40-50 &\sf y &\sf 65 + x + y \\\\\sf 50-60 &\sf 10 &\sf 75 + x + y\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{ \bf75 + x + y}&\frac{\qquad \qquad \qquad \qquad \qquad}{}\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\end{array}}$

___________________...

$\leadsto\sf 75 + x + y = 100$

$\leadsto\sf x + y = 100 - 75$

$\leadsto\sf x + y = 25 \: \: \: \: \Bigg\lgroup\bf{Equation (i)}\Bigg\rgroup$

______________________

$:\implies\sf N=\sum\limits f$

$:\implies\sf \dfrac{N}{2} = \dfrac{\sum\limits f}{2}$

$:\implies\sf \dfrac{N}{2} = \dfrac{100}{2}$

$:\implies\sf\dfrac{N}{2} = 50$

___________________

$\boxed{\begin{minipage}{6cm} \bigstar \:\:\sf Median = l + \sf\dfrac{\frac{n}{2}-C.f.}{f}\times h\\\\Here:\\1)\:l=Lower\:limit\:of\:median\:class=30\\2)\:C.f.=Cumulative\:frequency\:of\:class\\preceeding\:the\:median\:class=35 + x\\3)\:f= frequency\:of\:median\:class=30\\4)\:h= Class\:interval =10\end{minipage}}$

_________________________

$\underline{\bigstar\:\textbf{According to the Question :}}$

$\dashrightarrow\sf\:\:Median = l +\dfrac{\frac{n}{2}-C.f.}{f}\times h$

$\dashrightarrow\sf\:\:32 = 30 +\bigg\lgroup\dfrac{50-35 - x}{30}\bigg\rgroup \times 10$

$\dashrightarrow\sf\:\:32 - 30 = \dfrac{50 - 30 - x}{3}$

$\dashrightarrow\sf\:\:2 \times 3 = 15 - x$

$\dashrightarrow\sf\:\: x = 15 - 6$

$\dashrightarrow\:\: \underline{ \boxed{\textsf{ \textbf{ x = 9}}}}$

_____________________.....

$\qquad\tiny \dag\:\:\underline{\tt Putting \: value \: of \: x = 9 \: in\: equation \: (i) \: we \: get :}$

$\dashrightarrow\sf\:\:9 + y = 25$

$\dashrightarrow\sf\:\: y = 25 - 9$

$\dashrightarrow\:\: \underline{ \boxed{\textsf{ \textbf{ y = 16}}}}$