Ex.14.3 Q7 Statistics Solution - NCERT Maths Class 10

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Question

The distribution below gives the weights of \(30\) students of a class. Find the median weight of the students.

Weight (in kg) \(40-45\) \(45-50 \) \(50-55\) \(55-60\) \(60-65\) \(65-70\) \(70-75\)
Number of students \(2\) \(3\) \(8\) \(6\) \(6\) \(3\) \(2\)

  

Text Solution

What is known?

The weights of \(30\) students of a class.

What is unknown?

The median weight of the students.

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:

Weight (in kg)

Number of students

\(f\)

Cumulative frequency

\(cf\)

\(40-45\) \(2\) \(2\)
\(45-50 \) \(3\) \(2 + 3 = 5\)
\(50-55\) \(8\) \(5 + 8 = 13\)
\(55-60\) \(6\) \(13 + 6 = 19\)
\(60-65\) \(6\) \(19 + 6 = 25\)
\(65-70\) \(3\) \(25 + 3 = 2\)
\(70-75\) \(2\) \(28 + 2 = 30\)
\(n=30 \)

From the table, it can be observed that

\(n = 30{\rm{    }} \Rightarrow \frac{n}{2} = 15\)

Cumulative frequency \((cf)\) just greater than \(15\) is \(19,\) belonging to class \(55 – 60.\)

Therefore, median class \(=55 – 60\)

Class size\(, h = 5\)

Lower limit of median class\(, l = 55\)

Frequency of median class\(, f = 6\)

Cumulative frequency of class preceding median class, \(cf = 13\)

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

\[\begin{array}{l}
 = 55 + \left( {\frac{{15 - 13}}{6}} \right) \times 5\\
 = 55 + \frac{2}{6} \times 5\\
 = 55 + \frac{5}{3}\\
 = 55 + 1.67\\
 = 56.67
\end{array}\]

Therefore, median weight is \(56.67 \rm\,kg.\)

 

  
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