Question

Find the median of the following data:Class 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120 120 – 140Frequency 10 35 52 61 38 29

Answer 1

Answer:

65.08

Step-by-step explanation:

$\left(\begin{array}{ccc}C.I.&Frequency&cf\\0-20&10&10\\20-40&35&45\\40-50&52&97\\60-80&61&158\\80-100&38&196\\100-120&29&225\\&N=225&\end{array}\right)$

Now,

N/2 = 225/2 = 112.5

Since , the Commnulative frequency just greater than 112.5 is 158.

So, The class 60-80 is the median class,.

We have,

L= 60

cf = 97

h = 20

f = 61

Thus,

$\text{Median}=L+(\frac{\frac{n}{2}-cf}{f})\times h\\\;\\=60+(\frac{112.5-97}{61})\times20\\\;\\=65.08$

Answer 2

Answer:

$\begin{tabular}{|c|c|c|c|c|c|c|}\cline{1-7} Class & 0-20 & 20-40 & 40-60 & 60-80 &80-100&100-120\\\cline{1-7} Frequency&10&35&52&61&38&29\\\cline{1-7}\end{tabular}$

$\begin{tabular}{|c|c|c|}\cline{1-3}Class& Frequency & Cumulative Frequency\\\cline{1-3}0 - 20 & 10 & 10 \\\cline{1-3}20 - 40 & 35 & 45\\\cline{1-3}40 - 60 & 52 & 97\\\cline{1-3}60 - 80 & 61 & 158\\\cline{1-3}80 - 100 & 38 & 196\\\cline{1-3}100 - 120&29&225\\\cline{1-3}&\sum\limits f=225&\\\cline{1-3}\end{tabular}$

$\rule{130}{1}$

$\implies\sf N=\sum\limits f\\\\\\:\implies\sf \dfrac{N}{2} = \dfrac{\sum\limits f}{2}\\\\\\:\implies\sf \dfrac{N}{2} = \dfrac{225}{2}\\\\\\:\implies\sf\dfrac{N}{2} = 112.5 \\\\\underline{\textsf{Hence, 60 - 80 is the median class.}}$

$\boxed{\begin{minipage}{6cm} \bigstar \:\:\sf Median = l + \sf\dfrac{\frac{n}{2}-C.f.}{f}\times h\\\\Here:\\1)\:l=Lower\:limit\:of\:median\:class=60\\2)\:C.f.=Cumulative\:frequency\:of\:class\\preceeding\:the\:median\:class=97</p><p>\\3)\:f= frequency\:of\:median\:class=61</p><p>\\4)\:h= Class\:interval =80-60=20\end{minipage}}$

$\rule{190}{2}$

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

$\dashrightarrow\sf\:\:Median = l +\dfrac{\frac{n}{2}-C.f.}{f}\times h\\\\\\\dashrightarrow\sf\:\:Median = 60 +\bigg\lgroup\dfrac{112.5-97}{61}\times20\bigg\rgroup\\\\\\\dashrightarrow\sf\:\:Median = 60 +\bigg\lgroup\dfrac{15.5}{61}\times20\bigg\rgroup\\\\\\\dashrightarrow\sf\:\:Median = 60 +\dfrac{310}{61}\\\\\\\dashrightarrow\sf\:\:Median = 60 + 5.08\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf Median = 65.08}}$

⠀

$\therefore\:\underline{\textsf{Hence, Median of the given data is \textbf{65.08}}}.$