Find the median of :

(i) 15, 6, 16, 8, 22, 21, 9, 18, 25

(ii) 10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27

(iii) 55, 60, 35 ,51, 29, 63, 72, 91, 85, 82

(i) By arranging data in ascending order, we get :

6, 8, 9, 15, 16, 18, 21, 22, 25

Number of observations, n = 9, which is odd.

By formula,

Median = $\dfrac{n + 1}{2}$ th observation

⇒ $\dfrac{9 + 1}{2}$ th observation

⇒ $\dfrac{10}{2}$th observation

⇒ 5th observation

∴ Median = 16.

Hence, median is 16.

(ii) By arranging data in ascending order, we get :

3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81

Number of observations, n = 12, which is even.

By formula,

$\Rightarrow \text{Median} = \dfrac{\left(\dfrac{n}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{n}{2} + 1\right)^{\text{th}} \text{term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\left(\dfrac{12}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{12}{2} + 1\right)^{\text{th}} \text{term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{6th term} + \text{7th term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{15} + \text{17}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{32}}{2} \\[1em]$

∴ Median = 16.

Hence, median = 16.

(iii) By arranging data in ascending order, we get :

29, 35, 51, 55, 60, 63, 72, 82, 85, 91

Number of observations, n = 10, which is even.

By formula,

$\Rightarrow \text{Median} = \dfrac{\left(\dfrac{n}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{n}{2} + 1\right)^{\text{th}} \text{term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\left(\dfrac{10}{2}\right)^{\text{th}} \text{term} + \left(\dfrac{10}{2} + 1\right)^{\text{th}} \text{term}}{2}\\[1em] \Rightarrow \text{Median} = \dfrac{\text{5th term} + \text{6th term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\text{60} + \text{63}}{2}\\[1em] \Rightarrow \text{Median} = \dfrac{\text{123}}{2}$

∴ Median = 61.5.

Hence, median is 61.5.