Question

If a variable takes the discrete values $\alpha + 4, \alpha - \dfrac{7}{2}, \alpha - \dfrac{5}{2}, \alpha - 3, \alpha - 2, \alpha + \dfrac{1}{2}, \alpha - \dfrac{1}{2},\alpha + 5$ where $\alpha > 0$ . Then find the median of these values.
A. $\alpha - \dfrac{5}{4}$
B. $\alpha - \dfrac{1}{2}$
C. $\alpha - 2$
D. $\alpha + \dfrac{5}{4}$

Answer 1

Hint: First, rearrange the given values in the ascending or descending order. Then calculate the total number of values. If the total number of terms is odd, then consider the middle term as the median. If the total number of terms is even, then consider the average of the middle two terms as the median.

Formula Used:
When there are an odd number of values in a data set, then the median is a ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ term of the arranged data.
When there are even number of values in a data set, then the median is $\left( {\dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}term + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}term}}{2}} \right)$, where n is the number of observations.

Complete step by step solution:
The given set of values is $\alpha + 4, \alpha - \dfrac{7}{2}, \alpha - \dfrac{5}{2}, \alpha - 3, \alpha - 2, \alpha + \dfrac{1}{2}, \alpha - \dfrac{1}{2},\alpha + 5$, where $\alpha > 0$.
Let’s rearrange the above values of the data set in ascending order.
The ascending order is: $\alpha - \dfrac{7}{2}, \alpha - 3, \alpha - \dfrac{5}{2}, \alpha - 2, \alpha - \dfrac{1}{2}, \alpha + \dfrac{1}{2}, \alpha + 4,\alpha + 5$
The total number of values in the given set is: 8
Since there is an even number of values in a data set.
So, to find the median of the given data set apply the formula $\left( {\dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}} \text{term} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}} \text{term}}}{2}} \right)$.
We get,
$Median = \left( {\dfrac{{{{\left( {\dfrac{8}{2}} \right)}^{th}} \text{term} + {{\left( {\dfrac{8}{2} + 1} \right)}^{th}} \text{term}}}{2}} \right)$
$ \Rightarrow \text{Median} = \left( {\dfrac{{{4^{th}} \text{term} + {5^{th}} \text{term}}}{2}} \right)$
$ \Rightarrow \text{Median} = \dfrac{{\left( {\alpha - 2} \right) + \left( {\alpha - \dfrac{1}{2}} \right)}}{2}$
$ \Rightarrow \text{Median} = \dfrac{{\left( {2\alpha - \dfrac{5}{2}} \right)}}{2}$
$ \Rightarrow \text{Median} = \alpha - \dfrac{5}{4}$
Therefore, the median of the given discrete values is $\alpha - \dfrac{5}{4}$.

Option ‘A’ is correct

Note: The median is the middle value of the data when the data is arranged in an ascending or descending order. So, always rearrange the values of the data set in ascending or descending order.