Find the median of each of the following sets of numbers:

(i) 25, 6, 13, 20, 15, 8, 22, 9, 16, 21, 18

(ii) 15, 32, 41, 13, 51, 35, 0, 18, 56, 39, 37

(iii) 40, 31, 25, 36, 27, 38, 28, 35

(iv) 56, 81, 51, 42, 69, 85, 72, 35, 66, 92

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

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

6, 8, 9, 13, 15, 16, 18, 20, 21, 22, 25

Number of observations, n = 11, which is odd.

By formula,

$\Rightarrow \text{Median} = \dfrac{\text{n} + 1}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = \dfrac{11 + 1}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = \dfrac{12}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = 6 \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = 16.$

Hence, median = 16.

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

0, 13, 15, 18, 32, 35, 37, 39, 41, 51, 56

Number of observations, n = 11, which is odd.

By formula,

$\Rightarrow \text{Median} = \dfrac{\text{n} + 1}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = \dfrac{11 + 1}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = \dfrac{12}{2} \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = 6 \text{ th} \text{ observation} \\[1em] \Rightarrow \text{Median} = 35$

Hence, median = 35.

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

25, 27, 28, 31, 35, 36, 38, 40

Number of observations, n = 8, which is even.

By formula,

$\Rightarrow \text{Median} = \dfrac{\Big(\dfrac{\text{n}}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{\text{n}}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\Big(\dfrac{8}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{8}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{4\text{th} \text{ term} + (4 + 1)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{4\text{ th} \text{ term} + 5\text{ th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{31 + 35}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{66}{2} \\[1em] \Rightarrow \text{Median} = 33$

Hence, median = 33.

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

35, 42, 51, 56, 66, 69, 72, 81, 85, 92

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

By formula,

$\Rightarrow \text{Median} = \dfrac{\Big(\dfrac{\text{n}}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{\text{n}}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\Big(\dfrac{10}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{10}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{5\text{th} \text{ term} + (5 + 1)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{5\text{th} \text{ term} + 6\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{66 + 69}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{135}{2} \\[1em] \Rightarrow \text{Median} = 67.5$

Hence, median = 67.5.

(v) 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,

$\Rightarrow \text{Median} = \dfrac{\Big(\dfrac{\text{n}}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{\text{n}}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{\Big(\dfrac{12}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{12}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{6\text{th} \text{ term} + (6 + 1)\text{th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{6\text{ th} \text{ term} + 7\text{ th} \text{ term}}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{15 + 17}{2} \\[1em] \Rightarrow \text{Median} = \dfrac{32}{2} \\[1em] \Rightarrow \text{Median} = 16$

Hence, median = 16.