Question

o find the median for the following distribution table C-i 0-10,10-20,20-30,30-40,40-50 Fi=6,9,15,9,1

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Answer 1

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

Frequency distribution table is as follow:-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \small\boxed{\begin{array}{c |c|c} \tt{Classes} & \tt{Frequency}& \tt{c.f.} \\ \dfrac{\qquad\qquad}{ \sf 0-10} &\dfrac{\qquad\qquad}{ \sf 6} &\dfrac{\qquad\qquad}{ \sf 6}& \\ \dfrac{\qquad\qquad}{ \sf 10-20} &\dfrac{\qquad\qquad}{ \sf 9}&\dfrac{\qquad\qquad}{ \sf 15} & \\ \dfrac{\qquad\qquad}{ \sf 20-30} &\dfrac{\qquad\qquad}{ \sf 15}&\dfrac{\qquad\qquad}{ \sf 30} & \\ \dfrac{\qquad\qquad}{ \sf 30-40} &\dfrac{\qquad\qquad}{ \sf 9}&\dfrac{\qquad\qquad}{ \sf 39} & \\ \dfrac{\qquad\qquad}{ \sf 40-50} &\dfrac{\qquad\qquad}{ \sf 1}&\dfrac{\qquad\qquad}{ \sf 40} &\end{array}} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

We know,

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

Here,

• l denotes lower limit of median class

• h denotes width of median class

• f denotes frequency of median class

• cf denotes cumulative frequency of the class preceding the median class

• N denotes sum of frequency

So, from above calculations, we concluded that

N = 40

So, $\dfrac{N}{2}$ = 20

So, we get

• Median class = 20 - 30

• l = 20

• h = 10

• f = 15

• cf = 15

By substituting all the given values in the formula,

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

$\rm \: Median= 20 + \Bigg \{10 \times \dfrac{ \bigg( 20 - 15\bigg)}{15} \Bigg \}$

$\rm \: Median= 20 + \Bigg \{2\times \dfrac{ 5}{3} \Bigg \}$

$\rm \: Median= 20 + \frac{10}{3}$

$\rm \: Median= \frac{60 + 10}{3}$

$\rm \: Median= \frac{70}{3}$

$\rm\implies \:\boxed{ \rm{ \:\rm \: Median= \frac{70}{3} \: \approx \: 23.33 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. Mean Using Direct Method

$\boxed{ \rm{ \:Mean = \dfrac{ \sum f_i x_i}{ \sum f_i} \: }}$

2. Mean using Short Cut Method

$\boxed{ \rm{ \:Mean = A + \dfrac{ \sum f_i d_i}{ \sum f_i} \: }}$

3. Mean using Step Deviation Method

$\boxed{ \rm{ \:Mean = A + \dfrac{ \sum f_i u_i}{ \sum f_i} \times h \: }}$