Ex.14.3 Q7 Statistics Solution - NCERT Maths Class 10
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 \( = l + \left( {\frac{{\frac{n}{2} - cf}}{f}} \right) \times h\)
\[\begin{align}
& = 55 + \left( {\frac{{15 - 13}}{6}} \right) \times 5\\
& = 55 + \frac{2}{6} \times 5\\
&= 55 + \frac{5}{3}\\
&= 55 + 1.67\\
&= 56.67
\end{align}\]
Therefore, median weight is \(56.67 \rm\,kg.\)