Question

You have given median and you have to find the missing frequency.
Given : Median = 24
and answer comes out to be x =25.
CLASS INTERVAL. f
0 - 10 5
10 - 20 25
20 - 30 x
30 - 40 18
40 - 50 7
please provide the answer with full explanation and answer comes out to be x is equal to 25.​

Answer 1

Given:

$\begin{tabular}{|c|c|c|c|c|c|}\cline{1-2}Class Interval&Frequency\\\cline{1-2} 0-10&5\\\cline{1-2}10-20&25\\\cline{1-2}20-30&x\\\cline{1-2}30-40&18\\\cline{1-2}40-50&7\\\cline{1-2}\end{tabular}$

Median of given data,M= 24

To Find:

Value of x

Solution:

We know that,

• Median M of a grouped frequency table is given by

$\pink{\underline{\boxed{\bf{M=l+\Bigg(\frac{\frac{N}{2}-C'}{f}}\Bigg)\times h}}}}$

where,

l is lower limit of Median class

N is the sum of all frequencies

C' is cumulative frequency of the class preceding the median class

f is the frequency of median class

h is width of the median class

• The class which is consisting of median is called median class.

━━━━━━━━━━━━━━━━━━━━━

On adding column of cumulative frequency in given table, we get

$\begin{tabular}{|c|c|c|c|c|c|}\cline{1-3}Class Interval&Frequency&C.F.\\\cline{1-3} 0-10&5&5\\\cline{1-3}10-20&25&30\\\cline{1-3}20-30&x&30+x\\\cline{1-3}30-40&18&48+x\\\cline{1-3}40-50&7&55+x\\\cline{1-3}Total&55+x\\\cline{1-2}\end{tabular}$

Since, the median is 24

So, the median class will be 20-30

Here,

l= 20

N= 50+x

C'= 30

f= x

h is 10

Let the median of given data be M

So,

$\longrightarrow\rm{M=l+\Bigg(\dfrac{\frac{N}{2}-C'}{f}}\Bigg)\times h}$

$\longrightarrow\rm{24=20+\Bigg(\dfrac{\dfrac{55+x}{2}-30}{x}}\Bigg)\times10}$

$\longrightarrow\rm{24-20=\Bigg(\dfrac{\dfrac{55+x-60}{2}}{x}}\Bigg)\times10}$

$\longrightarrow\rm{4=\Bigg(\dfrac{\dfrac{x-5}{2}}{x}}\Bigg)\times10}$

$\longrightarrow\rm{4=\Bigg(\dfrac{x-5}{\cancel{2}x}}\Bigg)\times\cancel{10}}$

$\longrightarrow\rm{4=\Bigg(\dfrac{x-5}{x}}\Bigg)\times5}$

$\longrightarrow\rm{4=\dfrac{5(x-5)}{x}}$

$\longrightarrow\rm{4x=5(x-5)}$

$\longrightarrow\rm{4x=5x-25}$

$\longrightarrow\rm{5x-25=4x}$

$\longrightarrow\rm{5x-4x=25}$

$\longrightarrow\rm\green{x=25}$

Hence, the value of missing frequency is 25.

Answer 2

Given:

$$\begin{lgathered}\begin{tabular}{|c|c|c|c|c|c|}\cline{1-2}Class Interval&Frequency\\\cline{1-2} 0-10&5\\\cline{1-2}10-20&25\\\cline{1-2}20-30&x\\\cline{1-2}30-40&18\\\cline{1-2}40-50&7\\\cline{1-2}\end{tabular}\end{lgathered}$$

Median of given data,M= 24

To Find:

Value of x

Solution:

We know that,

Median M of a grouped frequency table is given by

$$\pink{\underline{\boxed{\bf{M=l+\Bigg(\frac{\frac{N}{2}-C'}{f}}\Bigg)\times h}}}}$$

where,

l is lower limit of Median class

N is the sum of all frequencies

C' is cumulative frequency of the class preceding the median class

f is the frequency of median class

h is width of the median class

The class which is consisting of median is called median class.

━━━━━━━━━━━━━━━━━━━━━

On adding column of cumulative frequency in given table, we get

$$\begin{lgathered}\begin{tabular}{|c|c|c|c|c|c|}\cline{1-3}Class Interval&Frequency&C.F.\\\cline{1-3} 0-10&5&5\\\cline{1-3}10-20&25&30\\\cline{1-3}20-30&x&30+x\\\cline{1-3}30-40&18&48+x\\\cline{1-3}40-50&7&55+x\\\cline{1-3}Total&55+x\\\cline{1-2}\end{tabular}\end{lgathered}$$

Since, the median is 24

So, the median class will be 20-30

Here,

l= 20

N= 50+x

C'= 30

f= x

h is 10

Let the median of given data be M

So,

$$\longrightarrow\rm{M=l+\Bigg(\dfrac{\frac{N}{2}-C'}{f}}\Bigg)\times

$$\longrightarrow\rm{4=\Bigg(\dfrac{\dfrac{x-5}{2}}{x}}\Bigg)\times10}$$

$$\longrightarrow\rm{4=\Bigg(\dfrac{x-5}{\cancel{2}x}}\Bigg)\times\cancel{10}}$$

$$\longrightarrow\rm{4=\Bigg(\dfrac{x-5}{x}}\Bigg)\times5}$$

$$\longrightarrow\rm{4=\dfrac{5(x-5)}{x}}$$

$$\longrightarrow\rm{4x=5(x-5)}$$

$$\longrightarrow\rm{4x=5x-25}$$

$$\longrightarrow\rm{5x-25=4x}$$

$$\longrightarrow\rm{5x-4x=25}$$

$$\longrightarrow\rm\green{x=25}$$

Hence, the value of missing frequency is 25.