Question

For a symmetrical distribution ${Q_1} = 25$, and ${Q_3} = 45$ (${Q_1}$, and ${Q_3}$ are the first and third quartiles), then find the median of the distribution.
A. $20$
B. $25$
C. $30$
D. $35$

Answer 1

Hint: In the given data, the values of the first and third quartiles of a symmetrical distribution are given. Then use the formula of the median of a symmetrical distribution that is consist of the first and third quartiles to reach the required answer.

Formula Used:
The median of a symmetrical distribution is: ${Q_2} = \dfrac{{{Q_1} + {Q_3}}}{2}$

Complete step by step solution:
Given:
The values of the first and third quartiles of a symmetrical distribution are ${Q_1} = 25$, and ${Q_3} = 45$ respectively.
Let’s calculate the median of the given distribution.
The value of second quartile is the median of the distribution.
Apply the formula of a median ${Q_2} = \dfrac{{{Q_1} + {Q_3}}}{2}$.
${Q_2} = \dfrac{{25 + 45}}{2}$
$ \Rightarrow {Q_2} = \dfrac{{70}}{2}$
$ \Rightarrow {Q_2} = 35$
Therefore, the median of the given symmetrical distribution is $35$.

Option ‘D’ is correct

Additional information
A symmetrical distribution occurs when variable values appear at regular intervals, and the mean, median, and mode frequently occur at the same point.

Note: Students often get confused about the formulas of the first, second and third quartiles. The second quartile is also called the median of the data set.
Following are the formula of the quartiles:
First quartile: ${Q_1} = \dfrac{1}{4}{\left( {n + 1} \right)^{th}}term$
Second quartile: ${Q_2} = \dfrac{1}{2}{\left( {n + 1} \right)^{th}}term$
Third quartile: ${Q_3} = \dfrac{3}{4}{\left( {n + 1} \right)^{th}}term$