Question

find the median of following data :

class interval :- 0-20 ,20-40,40-60,60-80,80-100,100-120,120-140

frequency:- 6,8,10,12,6,5,3​

Answer 1

★ Cumulative frequency table refer to attachment ★

To find :-

• Median of given data

Solution :-

As we know that

→ Median = l + {h × (N/2 - cf/f)}

Where

• l = lower limit
• h = width of median class
• cf = cumulative frequency
• N = Σf

Now, according to given condition

→ N = Σf

→ N = 50

→ N/2 = 50/2 = 25

Cumulative frequency i.e greater than 25 is 36

∴ Median Class = 60 - 80

• l = 60
• h = 20
• f = 12
• c.f (preceding class) = 24

→ Me = l + {h × (N/2 - cf/f)}

Substitute all the values

→ Me = 60 + {20 × (25 - 24)/12}

→ Me = 60 + {20 × 1/12}

→ Me = 60 + 20/12

→ Me = 60 + 5/3

→ Me = 60 + 1.66

→ Me = 61.66

Hence,

• Median of given data is 61.66

Note :

• Me denotes Median
• Σ → Sigma → Summation

Answer 2

Answer:

$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency&\sf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0-20&\sf 6&\sf 6 \\\\\sf 20-40 &\sf 8&\sf 14 \\\\\sf 40-60 &\sf 10 &\sf 24 \\\\\sf 60-80&\sf 12&\sf 36\\\\\sf 80-100 &\sf 6 &\sf 42 \\\\\sf 100-120 &\sf 5 &\sf 47 \\\\\sf 120-140 &\sf 3 &\sf 50\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{\sum\limits\:f=50}}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\end{array}}$

________________________

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

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

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

$:\implies\sf\dfrac{N}{2} = 25$

$\underline{\textsf{Hence, 60 -80 is the median class.}}$

___________________

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

$\sf Here:$

$\sf 1) \:l=Lower\:limit\:of\:median\:class=60$

$\sf 2)\:C.f.=Cumulative\:frequency\:of\:class \: preceeding\:the\:median\:class=24$

$\sf 3)\:f= frequency\:of\:median\:class=12$

$\sf 4)\:h= Class\:interval = 20$

_____________________

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

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

$\dashrightarrow\sf\:\:Median = 60 +\bigg\lgroup\dfrac{25 - 24}{12}\times 20\bigg\rgroup$

$\dashrightarrow\sf\:\:Median = 60 +\bigg\lgroup\dfrac{1}{12}\times 20\bigg\rgroup$

$\dashrightarrow\sf\:\:Median = 60 +\dfrac{20}{12}$

$\dashrightarrow\sf\:\:Median = 60 + 1.66$

$\dashrightarrow\:\:\underline{\boxed{\sf Median = 61.66}}$

⠀

$\therefore\:\underline{\textsf{Median Height of the distribution is \textbf{61.66}}}.$