Ex.14.3 Q2 Statistics Solution - NCERT Maths Class 10

Go back to  'Ex.14.3'

Question

If the median of the distribution given below is \(28.5\), find the values of \(x\) and \(y.\)

Class interval Frequency
\(0 – 10\) \(5\)
\(10 – 20\) \(x\)
\(20 – 30\) \(20\)
\(30 – 40\) \(15\)
\(40 – 50\) \(y\)
\( 50 – 60\) \(5\)
Total  \(60\)

   

Text Solution

What is known?

The median of the distribution is \(28.5\)

What is the unknown?

The values of \(x\) and \(y.\)

Reasoning:

Median Class is the class having Cumulative frequency \((cf)\) just greater than \(\frac {n}2\)

Median \( = l + \left( {\frac{{\frac{n}{2} - cf}}{f}} \right) \times h\)

Class size,\(h\)

Number of observations,\(n\)

Lower limit of median class,\(l\)

Frequency of median class,\(f\)

Cumulative frequency of class preceding median class,\(cf\)

Steps:

The cumulative frequency for the given data is calculated as follows.

Class interval  Frequency Frequency
\(0 – 10\) \(5\) \(5\)
\(20 – 30\) \(x\) \(5 + x\)
\(30 – 40\) \(20\) \(25 + x\)
\(40 – 50\) \(15\) \(40 + x\)
\(40 – 50\) \(y\) \(40 + x + y\)
\( 50 – 60\) \(5\) \(45 +x+y\)
\(n=\) \(60\)

From the table, it can be observed that

\(n =60\)    \(\Rightarrow \frac{n}{2}=30\)

\[\begin{align} \\ {45+x+y}&={60} \\ {x+y} &={15}.............(i) \end{align}\]
Median of the data is given as \(28.5\) which lies in interval \(20 - 30.\)

Therefore, median class \(= 20 - 30\)

Class size (\(h\)) \(= 10\)

Lower limit of median class (\(l\)\(=20\)

Frequency of median class (\(f\)\(=20\)

Cumulative frequency of class preceding the median class, (\(cf\)\(=5 + x\)

\[\begin{align}{\text { Median }}&={l+\left(\frac{\frac{n}{2}-c f}{f}\right) \times h} \\ {28.5}&={20+\left(\frac{\frac{60}{2}-(5+x)}{20}\right) \times 10} \\ {8.5}&={\left(\frac{25-x}{2}\right)} \\ {17}&={25-x} \\ {x}&={8}\end{align}\]

Putting \(x=8\) equation, (i)

\[\begin{align} {8+y}&={15} \\ {y}&={7}\end{align}\]

Hence, the values of \(x\) and \(y\) are \(8\) and \(7\) respectively.

  
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school